__init__.py

'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):
'''This module implements specialized container datatypes providing
alternatives to Python's general purpose built-in containers, dict,
list, set, and tuple.

* namedtuple factory function for creating tuple subclasses with named fields
* deque list-like container with fast appends and pops on either end
* ChainMap dict-like class for creating a single view of multiple mappings
* Counter dict subclass for counting hashable objects
* OrderedDict dict subclass that remembers the order entries were added
* defaultdict dict subclass that calls a factory function to supply missing values
* UserDict wrapper around dictionary objects for easier dict subclassing
* UserList wrapper around list objects for easier list subclassing
* UserString wrapper around string objects for easier string subclassing

'''

__all__= [
'ChainMap',
'Counter',
'OrderedDict',
'UserDict',
'UserList',
'UserString',
'defaultdict',
'deque',
'namedtuple',
]

import_collections_abc
importsysas_sys

fromitertoolsimportchainas_chain
fromitertoolsimportrepeatas_repeat
fromitertoolsimportstarmapas_starmap
fromkeywordimportiskeywordas_iskeyword
fromoperatorimporteqas_eq
fromoperatorimportitemgetteras_itemgetter
fromreprlibimportrecursive_repras_recursive_repr
from_weakrefimportproxyas_proxy

try:
from_collectionsimportdeque
exceptImportError:
pass
else:
_collections_abc.MutableSequence.register(deque)

try:
from_collectionsimportdefaultdict
exceptImportError:
pass


################################################################################
### OrderedDict
################################################################################

class_OrderedDictKeysView(_collections_abc.KeysView):

def__reversed__(self):
yieldfromreversed(self._mapping)

class_OrderedDictItemsView(_collections_abc.ItemsView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yield (key, self._mapping[key])

class_OrderedDictValuesView(_collections_abc.ValuesView):

def__reversed__(self):
forkeyinreversed(self._mapping):
yieldself._mapping[key]

class_Link(object):
__slots__='prev', 'next', 'key', '__weakref__'

classOrderedDict(dict):
'Dictionary that remembers insertion order'
# An inherited dict maps keys to values.
# The inherited dict provides __getitem__, __len__, __contains__, and get.
# The remaining methods are order-aware.
# Big-O running times for all methods are the same as regular dictionaries.

# The internal self.__map dict maps keys to links in a doubly linked list.
# The circular doubly linked list starts and ends with a sentinel element.
# The sentinel element never gets deleted (this simplifies the algorithm).
# The sentinel is in self.__hardroot with a weakref proxy in self.__root.
# The prev links are weakref proxies (to prevent circular references).
# Individual links are kept alive by the hard reference in self.__map.
# Those hard references disappear when a key is deleted from an OrderedDict.

def__new__(cls, /, *args, **kwds):
"Create the ordered dict object and set up the underlying structures."
self=dict.__new__(cls)
self.__hardroot=_Link()
self.__root=root=_proxy(self.__hardroot)
root.prev=root.next=root
self.__map= {}
returnself

def__init__(self, other=(), /, **kwds):
'''Initialize an ordered dictionary. The signature is the same as
 regular dictionaries. Keyword argument order is preserved.
 '''
self.__update(other, **kwds)

def__setitem__(self, key, value,
dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
'od.__setitem__(i, y) <==> od[i]=y'
# Setting a new item creates a new link at the end of the linked list,
# and the inherited dictionary is updated with the new key/value pair.
ifkeynotinself:
self.__map[key] =link=Link()
root=self.__root
last=root.prev
link.prev, link.next, link.key=last, root, key
last.next=link
root.prev=proxy(link)
dict_setitem(self, key, value)

def__delitem__(self, key, dict_delitem=dict.__delitem__):
'od.__delitem__(y) <==> del od[y]'
# Deleting an existing item uses self.__map to find the link which gets
# removed by updating the links in the predecessor and successor nodes.
dict_delitem(self, key)
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None

def__iter__(self):
'od.__iter__() <==> iter(od)'
# Traverse the linked list in order.
root=self.__root
curr=root.next
whilecurrisnotroot:
yieldcurr.key
curr=curr.next

def__reversed__(self):
'od.__reversed__() <==> reversed(od)'
# Traverse the linked list in reverse order.
root=self.__root
curr=root.prev
whilecurrisnotroot:
yieldcurr.key
curr=curr.prev

defclear(self):
'od.clear() -> None. Remove all items from od.'
root=self.__root
root.prev=root.next=root
self.__map.clear()
dict.clear(self)

defpopitem(self, last=True):
'''Remove and return a (key, value) pair from the dictionary.

 Pairs are returned in LIFO order if last is true or FIFO order if false.
 '''
ifnotself:
raiseKeyError('dictionary is empty')
root=self.__root
iflast:
link=root.prev
link_prev=link.prev
link_prev.next=root
root.prev=link_prev
else:
link=root.next
link_next=link.next
root.next=link_next
link_next.prev=root
key=link.key
delself.__map[key]
value=dict.pop(self, key)
returnkey, value

defmove_to_end(self, key, last=True):
'''Move an existing element to the end (or beginning if last is false).

 Raise KeyError if the element does not exist.
 '''
link=self.__map[key]
link_prev=link.prev
link_next=link.next
soft_link=link_next.prev
link_prev.next=link_next
link_next.prev=link_prev
root=self.__root
iflast:
last=root.prev
link.prev=last
link.next=root
root.prev=soft_link
last.next=link
else:
first=root.next
link.prev=root
link.next=first
first.prev=soft_link
root.next=link

def__sizeof__(self):
sizeof=_sys.getsizeof
n=len(self) +1# number of links including root
size=sizeof(self.__dict__) # instance dictionary
size+=sizeof(self.__map) *2# internal dict and inherited dict
size+=sizeof(self.__hardroot) *n# link objects
size+=sizeof(self.__root) *n# proxy objects
returnsize

update=__update=_collections_abc.MutableMapping.update

defkeys(self):
"D.keys() -> a set-like object providing a view on D's keys"
return_OrderedDictKeysView(self)

defitems(self):
"D.items() -> a set-like object providing a view on D's items"
return_OrderedDictItemsView(self)

defvalues(self):
"D.values() -> an object providing a view on D's values"
return_OrderedDictValuesView(self)

__ne__=_collections_abc.MutableMapping.__ne__

__marker=object()

defpop(self, key, default=__marker):
'''od.pop(k[,d]) -> v, remove specified key and return the corresponding
 value. If key is not found, d is returned if given, otherwise KeyError
 is raised.

 '''
marker=self.__marker
result=dict.pop(self, key, marker)
ifresultisnotmarker:
# The same as in __delitem__().
link=self.__map.pop(key)
link_prev=link.prev
link_next=link.next
link_prev.next=link_next
link_next.prev=link_prev
link.prev=None
link.next=None
returnresult
ifdefaultismarker:
raiseKeyError(key)
returndefault

defsetdefault(self, key, default=None):
'''Insert key with a value of default if key is not in the dictionary.

 Return the value for key if key is in the dictionary, else default.
 '''
ifkeyinself:
returnself[key]
self[key] =default
returndefault

@_recursive_repr()
def__repr__(self):
'od.__repr__() <==> repr(od)'
ifnotself:
return'%s()'% (self.__class__.__name__,)
return'%s(%r)'% (self.__class__.__name__, list(self.items()))

def__reduce__(self):
'Return state information for pickling'
state=self.__getstate__()
ifstate:
ifisinstance(state, tuple):
state, slots=state
else:
slots= {}
state=state.copy()
slots=slots.copy()
forkinvars(OrderedDict()):
state.pop(k, None)
slots.pop(k, None)
ifslots:
state=state, slots
else:
state=stateorNone
returnself.__class__, (), state, None, iter(self.items())

defcopy(self):
'od.copy() -> a shallow copy of od'
returnself.__class__(self)

@classmethod
deffromkeys(cls, iterable, value=None):
'''Create a new ordered dictionary with keys from iterable and values set to value.
 '''
self=cls()
forkeyiniterable:
self[key] =value
returnself

def__eq__(self, other):
'''od.__eq__(y) <==> od==y. Comparison to another OD is order-sensitive
 while comparison to a regular mapping is order-insensitive.

 '''
ifisinstance(other, OrderedDict):
returndict.__eq__(self, other) andall(map(_eq, self, other))
returndict.__eq__(self, other)

def__ior__(self, other):
self.update(other)
returnself

def__or__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(self)
new.update(other)
returnnew

def__ror__(self, other):
ifnotisinstance(other, dict):
returnNotImplemented
new=self.__class__(other)
new.update(self)
returnnew


try:
from_collectionsimportOrderedDict
exceptImportError:
# Leave the pure Python version in place.
pass


################################################################################
### namedtuple
################################################################################

try:
from_collectionsimport_tuplegetter
exceptImportError:
_tuplegetter=lambdaindex, doc: property(_itemgetter(index), doc=doc)

defnamedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
"""Returns a new subclass of tuple with named fields.

 >>> Point = namedtuple('Point', ['x', 'y'])
 >>> Point.__doc__ # docstring for the new class
 'Point(x, y)'
 >>> p = Point(11, y=22) # instantiate with positional args or keywords
 >>> p[0] + p[1] # indexable like a plain tuple
 33
 >>> x, y = p # unpack like a regular tuple
 >>> x, y
 (11, 22)
 >>> p.x + p.y # fields also accessible by name
 33
 >>> d = p._asdict() # convert to a dictionary
 >>> d['x']
 11
 >>> Point(**d) # convert from a dictionary
 Point(x=11, y=22)
 >>> p._replace(x=100) # _replace() is like str.replace() but targets named fields
 Point(x=100, y=22)

 """

# Validate the field names. At the user's option, either generate an error
# message or automatically replace the field name with a valid name.
ifisinstance(field_names, str):
field_names=field_names.replace(',', ' ').split()
field_names=list(map(str, field_names))
typename=_sys.intern(str(typename))

ifrename:
seen=set()
forindex, nameinenumerate(field_names):
if (notname.isidentifier()
or_iskeyword(name)
orname.startswith('_')
ornameinseen):
field_names[index] =f'_{index}'
seen.add(name)

fornamein [typename] +field_names:
iftype(name) isnotstr:
raiseTypeError('Type names and field names must be strings')
ifnotname.isidentifier():
raiseValueError('Type names and field names must be valid '
f'identifiers: {name!r}')
if_iskeyword(name):
raiseValueError('Type names and field names cannot be a '
f'keyword: {name!r}')

seen=set()
fornameinfield_names:
ifname.startswith('_') andnotrename:
raiseValueError('Field names cannot start with an underscore: '
f'{name!r}')
ifnameinseen:
raiseValueError(f'Encountered duplicate field name: {name!r}')
seen.add(name)

field_defaults= {}
ifdefaultsisnotNone:
defaults=tuple(defaults)
iflen(defaults) >len(field_names):
raiseTypeError('Got more default values than field names')
field_defaults=dict(reversed(list(zip(reversed(field_names),
reversed(defaults)))))

# Variables used in the methods and docstrings
field_names=tuple(map(_sys.intern, field_names))
num_fields=len(field_names)
arg_list=', '.join(field_names)
ifnum_fields==1:
arg_list+=','
repr_fmt='('+', '.join(f'{name}=%r'fornameinfield_names) +')'
tuple_new=tuple.__new__
_dict, _tuple, _len, _map, _zip=dict, tuple, len, map, zip

# Create all the named tuple methods to be added to the class namespace

namespace= {
'_tuple_new': tuple_new,
'__builtins__': {},
'__name__': f'namedtuple_{typename}',
 }
code=f'lambda _cls, {arg_list}: _tuple_new(_cls, ({arg_list}))'
__new__=eval(code, namespace)
__new__.__name__='__new__'
__new__.__doc__=f'Create new instance of {typename}({arg_list})'
ifdefaultsisnotNone:
__new__.__defaults__=defaults

@classmethod
def_make(cls, iterable):
result=tuple_new(cls, iterable)
if_len(result) !=num_fields:
raiseTypeError(f'Expected {num_fields} arguments, got {len(result)}')
returnresult

_make.__func__.__doc__= (f'Make a new {typename} object from a sequence '
'or iterable')

def_replace(self, /, **kwds):
result=self._make(_map(kwds.pop, field_names, self))
ifkwds:
raiseValueError(f'Got unexpected field names: {list(kwds)!r}')
returnresult

_replace.__doc__= (f'Return a new {typename} object replacing specified '
'fields with new values')

def__repr__(self):
'Return a nicely formatted representation string'
returnself.__class__.__name__+repr_fmt%self

def_asdict(self):
'Return a new dict which maps field names to their values.'
return_dict(_zip(self._fields, self))

def__getnewargs__(self):
'Return self as a plain tuple. Used by copy and pickle.'
return_tuple(self)

# Modify function metadata to help with introspection and debugging
formethodin (
__new__,
_make.__func__,
_replace,
__repr__,
_asdict,
__getnewargs__,
 ):
method.__qualname__=f'{typename}.{method.__name__}'

# Build-up the class namespace dictionary
# and use type() to build the result class
class_namespace= {
'__doc__': f'{typename}({arg_list})',
'__slots__': (),
'_fields': field_names,
'_field_defaults': field_defaults,
'__new__': __new__,
'_make': _make,
'_replace': _replace,
'__repr__': __repr__,
'_asdict': _asdict,
'__getnewargs__': __getnewargs__,
'__match_args__': field_names,
 }
forindex, nameinenumerate(field_names):
doc=_sys.intern(f'Alias for field number {index}')
class_namespace[name] =_tuplegetter(index, doc)

result=type(typename, (tuple,), class_namespace)

# For pickling to work, the __module__ variable needs to be set to the frame
# where the named tuple is created. Bypass this step in environments where
# sys._getframe is not defined (Jython for example) or sys._getframe is not
# defined for arguments greater than 0 (IronPython), or where the user has
# specified a particular module.
ifmoduleisNone:
try:
module=_sys._getframe(1).f_globals.get('__name__', '__main__')
except (AttributeError, ValueError):
pass
ifmoduleisnotNone:
result.__module__=module

returnresult


########################################################################
### Counter
########################################################################

def_count_elements(mapping, iterable):
'Tally elements from the iterable.'
mapping_get=mapping.get
foreleminiterable:
mapping[elem] =mapping_get(elem, 0) +1

try: # Load C helper function if available
from_collectionsimport_count_elements
exceptImportError:
pass

classCounter(dict):
'''Dict subclass for counting hashable items. Sometimes called a bag
 or multiset. Elements are stored as dictionary keys and their counts
 are stored as dictionary values.

 >>> c = Counter('abcdeabcdabcaba') # count elements from a string

 >>> c.most_common(3) # three most common elements
 [('a', 5), ('b', 4), ('c', 3)]
 >>> sorted(c) # list all unique elements
 ['a', 'b', 'c', 'd', 'e']
 >>> ''.join(sorted(c.elements())) # list elements with repetitions
 'aaaaabbbbcccdde'
 >>> sum(c.values()) # total of all counts
 15

 >>> c['a'] # count of letter 'a'
 5
 >>> for elem in 'shazam': # update counts from an iterable
 ... c[elem] += 1 # by adding 1 to each element's count
 >>> c['a'] # now there are seven 'a'
 7
 >>> del c['b'] # remove all 'b'
 >>> c['b'] # now there are zero 'b'
 0

 >>> d = Counter('simsalabim') # make another counter
 >>> c.update(d) # add in the second counter
 >>> c['a'] # now there are nine 'a'
 9

 >>> c.clear() # empty the counter
 >>> c
 Counter()

 Note: If a count is set to zero or reduced to zero, it will remain
 in the counter until the entry is deleted or the counter is cleared:

 >>> c = Counter('aaabbc')
 >>> c['b'] -= 2 # reduce the count of 'b' by two
 >>> c.most_common() # 'b' is still in, but its count is zero
 [('a', 3), ('c', 1), ('b', 0)]

 '''
# References:
# http://en.wikipedia.org/wiki/Multiset
# http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
# http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
# http://code.activestate.com/recipes/259174/
# Knuth, TAOCP Vol. II section 4.6.3

def__init__(self, iterable=None, /, **kwds):
'''Create a new, empty Counter object. And if given, count elements
 from an input iterable. Or, initialize the count from another mapping
 of elements to their counts.

 >>> c = Counter() # a new, empty counter
 >>> c = Counter('gallahad') # a new counter from an iterable
 >>> c = Counter({'a': 4, 'b': 2}) # a new counter from a mapping
 >>> c = Counter(a=4, b=2) # a new counter from keyword args

 '''
super().__init__()
self.update(iterable, **kwds)

def__missing__(self, key):
'The count of elements not in the Counter is zero.'
# Needed so that self[missing_item] does not raise KeyError
return0

deftotal(self):
'Sum of the counts'
returnsum(self.values())

defmost_common(self, n=None):
'''List the n most common elements and their counts from the most
 common to the least. If n is None, then list all element counts.

 >>> Counter('abracadabra').most_common(3)
 [('a', 5), ('b', 2), ('r', 2)]

 '''
# Emulate Bag.sortedByCount from Smalltalk
ifnisNone:
returnsorted(self.items(), key=_itemgetter(1), reverse=True)

# Lazy import to speedup Python startup time
importheapq
returnheapq.nlargest(n, self.items(), key=_itemgetter(1))

defelements(self):
'''Iterator over elements repeating each as many times as its count.

 >>> c = Counter('ABCABC')
 >>> sorted(c.elements())
 ['A', 'A', 'B', 'B', 'C', 'C']

 # Knuth's example for prime factors of 1836: 2**2 * 3**3 * 17**1
 >>> import math
 >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
 >>> math.prod(prime_factors.elements())
 1836

 Note, if an element's count has been set to zero or is a negative
 number, elements() will ignore it.

 '''
# Emulate Bag.do from Smalltalk and Multiset.begin from C++.
return_chain.from_iterable(_starmap(_repeat, self.items()))

# Override dict methods where necessary

@classmethod
deffromkeys(cls, iterable, v=None):
# There is no equivalent method for counters because the semantics
# would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
# Initializing counters to zero values isn't necessary because zero
# is already the default value for counter lookups. Initializing
# to one is easily accomplished with Counter(set(iterable)). For
# more exotic cases, create a dictionary first using a dictionary
# comprehension or dict.fromkeys().
raiseNotImplementedError(
'Counter.fromkeys() is undefined. Use Counter(iterable) instead.')

defupdate(self, iterable=None, /, **kwds):
'''Like dict.update() but add counts instead of replacing them.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.update('witch') # add elements from another iterable
 >>> d = Counter('watch')
 >>> c.update(d) # add elements from another counter
 >>> c['h'] # four 'h' in which, witch, and watch
 4

 '''
# The regular dict.update() operation makes no sense here because the
# replace behavior results in some of the original untouched counts
# being mixed-in with all of the other counts for a mismash that
# doesn't have a straight-forward interpretation in most counting
# contexts. Instead, we implement straight-addition. Both the inputs
# and outputs are allowed to contain zero and negative counts.

ifiterableisnotNone:
ifisinstance(iterable, _collections_abc.Mapping):
ifself:
self_get=self.get
forelem, countiniterable.items():
self[elem] =count+self_get(elem, 0)
else:
# fast path when counter is empty
super().update(iterable)
else:
_count_elements(self, iterable)
ifkwds:
self.update(kwds)

defsubtract(self, iterable=None, /, **kwds):
'''Like dict.update() but subtracts counts instead of replacing them.
 Counts can be reduced below zero. Both the inputs and outputs are
 allowed to contain zero and negative counts.

 Source can be an iterable, a dictionary, or another Counter instance.

 >>> c = Counter('which')
 >>> c.subtract('witch') # subtract elements from another iterable
 >>> c.subtract(Counter('watch')) # subtract elements from another counter
 >>> c['h'] # 2 in which, minus 1 in witch, minus 1 in watch
 0
 >>> c['w'] # 1 in which, minus 1 in witch, minus 1 in watch
 -1

 '''
ifiterableisnotNone:
self_get=self.get
ifisinstance(iterable, _collections_abc.Mapping):
forelem, countiniterable.items():
self[elem] =self_get(elem, 0) -count
else:
foreleminiterable:
self[elem] =self_get(elem, 0) -1
ifkwds:
self.subtract(kwds)

defcopy(self):
'Return a shallow copy.'
returnself.__class__(self)

def__reduce__(self):
returnself.__class__, (dict(self),)

def__delitem__(self, elem):
'Like dict.__delitem__() but does not raise KeyError for missing values.'
ifeleminself:
super().__delitem__(elem)

def__repr__(self):
ifnotself:
returnf'{self.__class__.__name__}()'
try:
# dict() preserves the ordering returned by most_common()
d=dict(self.most_common())
exceptTypeError:
# handle case where values are not orderable
d=dict(self)
returnf'{self.__class__.__name__}({d!r})'

# Multiset-style mathematical operations discussed in:
# Knuth TAOCP Volume II section 4.6.3 exercise 19
# and at http://en.wikipedia.org/wiki/Multiset
#
# Outputs guaranteed to only include positive counts.
#
# To strip negative and zero counts, add-in an empty counter:
# c += Counter()
#
# Results are ordered according to when an element is first
# encountered in the left operand and then by the order
# encountered in the right operand.
#
# When the multiplicities are all zero or one, multiset operations
# are guaranteed to be equivalent to the corresponding operations
# for regular sets.
# Given counter multisets such as:
# cp = Counter(a=1, b=0, c=1)
# cq = Counter(c=1, d=0, e=1)
# The corresponding regular sets would be:
# sp = {'a', 'c'}
# sq = {'c', 'e'}
# All of the following relations would hold:
# set(cp + cq) == sp | sq
# set(cp - cq) == sp - sq
# set(cp | cq) == sp | sq
# set(cp & cq) == sp & sq
# (cp == cq) == (sp == sq)
# (cp != cq) == (sp != sq)
# (cp <= cq) == (sp <= sq)
# (cp < cq) == (sp < sq)
# (cp >= cq) == (sp >= sq)
# (cp > cq) == (sp > sq)

def__eq__(self, other):
'True if all counts agree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] ==other[e] forcin (self, other) foreinc)

def__ne__(self, other):
'True if any counts disagree. Missing counts are treated as zero.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnnotself==other

def__le__(self, other):
'True if all counts in self are a subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] <=other[e] forcin (self, other) foreinc)

def__lt__(self, other):
'True if all counts in self are a proper subset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself<=otherandself!=other

def__ge__(self, other):
'True if all counts in self are a superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnall(self[e] >=other[e] forcin (self, other) foreinc)

def__gt__(self, other):
'True if all counts in self are a proper superset of those in other.'
ifnotisinstance(other, Counter):
returnNotImplemented
returnself>=otherandself!=other

def__add__(self, other):
'''Add counts from two counters.

 >>> Counter('abbb') + Counter('bcc')
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count+other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__sub__(self, other):
''' Subtract count, but keep only results with positive counts.

 >>> Counter('abbbc') - Counter('bccd')
 Counter({'b': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
newcount=count-other[elem]
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount<0:
result[elem] =0-count
returnresult

def__or__(self, other):
'''Union is the maximum of value in either of the input counters.

 >>> Counter('abbb') | Counter('bcc')
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=other_countifcount<other_countelsecount
ifnewcount>0:
result[elem] =newcount
forelem, countinother.items():
ifelemnotinselfandcount>0:
result[elem] =count
returnresult

def__and__(self, other):
''' Intersection is the minimum of corresponding counts.

 >>> Counter('abbb') & Counter('bcc')
 Counter({'b': 1})

 '''
ifnotisinstance(other, Counter):
returnNotImplemented
result=Counter()
forelem, countinself.items():
other_count=other[elem]
newcount=countifcount<other_countelseother_count
ifnewcount>0:
result[elem] =newcount
returnresult

def__pos__(self):
'Adds an empty counter, effectively stripping negative and zero counts'
result=Counter()
forelem, countinself.items():
ifcount>0:
result[elem] =count
returnresult

def__neg__(self):
'''Subtracts from an empty counter. Strips positive and zero counts,
 and flips the sign on negative counts.

 '''
result=Counter()
forelem, countinself.items():
ifcount<0:
result[elem] =0-count
returnresult

def_keep_positive(self):
'''Internal method to strip elements with a negative or zero count'''
nonpositive= [elemforelem, countinself.items() ifnotcount>0]
foreleminnonpositive:
delself[elem]
returnself

def__iadd__(self, other):
'''Inplace add from another counter, keeping only positive counts.

 >>> c = Counter('abbb')
 >>> c += Counter('bcc')
 >>> c
 Counter({'b': 4, 'c': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] +=count
returnself._keep_positive()

def__isub__(self, other):
'''Inplace subtract counter, but keep only results with positive counts.

 >>> c = Counter('abbbc')
 >>> c -= Counter('bccd')
 >>> c
 Counter({'b': 2, 'a': 1})

 '''
forelem, countinother.items():
self[elem] -=count
returnself._keep_positive()

def__ior__(self, other):
'''Inplace union is the maximum of value from either counter.

 >>> c = Counter('abbb')
 >>> c |= Counter('bcc')
 >>> c
 Counter({'b': 3, 'c': 2, 'a': 1})

 '''
forelem, other_countinother.items():
count=self[elem]
ifother_count>count:
self[elem] =other_count
returnself._keep_positive()

def__iand__(self, other):
'''Inplace intersection is the minimum of corresponding counts.

 >>> c = Counter('abbb')
 >>> c &= Counter('bcc')
 >>> c
 Counter({'b': 1})

 '''
forelem, countinself.items():
other_count=other[elem]
ifother_count<count:
self[elem] =other_count
returnself._keep_positive()


########################################################################
### ChainMap
########################################################################

classChainMap(_collections_abc.MutableMapping):
''' A ChainMap groups multiple dicts (or other mappings) together
 to create a single, updateable view.

 The underlying mappings are stored in a list. That list is public and can
 be accessed or updated using the *maps* attribute. There is no other
 state.

 Lookups search the underlying mappings successively until a key is found.
 In contrast, writes, updates, and deletions only operate on the first
 mapping.

 '''

def__init__(self, *maps):
'''Initialize a ChainMap by setting *maps* to the given mappings.
 If no mappings are provided, a single empty dictionary is used.

 '''
self.maps=list(maps) or [{}] # always at least one map

def__missing__(self, key):
raiseKeyError(key)

def__getitem__(self, key):