Dynamic programming/matrix chain multiplication#10549

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Dynamic programming/matrix chain multiplication #10549

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new algorithm matrix_chain_multiplication
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4 changes: 3 additions & 1 deletion DIRECTORY.md
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Expand Up@@ -265,6 +265,7 @@
* [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
* [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
* [Stack](data_structures/stacks/stack.py)
* [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
* [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
* [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
* [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
* [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
* [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
* [Longest Sub Array](dynamic_programming/longest_sub_array.py)
* [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
* [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
* [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
* [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
* [Altitude Pressure](physics/altitude_pressure.py)
* [Archimedes Principle](physics/archimedes_principle.py)
* [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
* [Basic Orbital Capture](physics/basic_orbital_capture.py)
* [Casimir Effect](physics/casimir_effect.py)
* [Centripetal Force](physics/centripetal_force.py)
Expand Down
78 changes: 78 additions & 0 deletions dynamic_programming/matrix_chain_multiplication.py
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"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
"""
Find the minimum number of multiplcations to multiply
chain of matrices.

Args:
arr : The input array of integers.

Returns:
int: Minimum number of multiplications needed to multiply the chain

Examples:
>>> matrix_chain_multiply([1,2,3,4,3])
30
>>> matrix_chain_multiply([10])
0
>>> matrix_chain_multiply([10, 20])
0
>>> matrix_chain_multiply([19, 2, 19])
722
"""
# first edge case
if len(arr) < 2:
return 0
# initialising 2D dp matrix
n = len(arr)
dp = [[float("inf") for j in range(n)] for i in range(n)]
# we want minimum cost of multiplication of matrices
# of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
for i in range(n - 1, 0, -1):
for j in range(i, n):
if i == j:
dp[i][j] = 0
continue
for k in range(i, j):
dp[i][j] = min(
dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
)
return dp[1][n - 1]


if __name__ == "__main__":
import doctest

doctest.testmod()
, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
 blocks\n(function() {\n function addCopyButtons() {\n document.querySelectorAll('pre code').forEach(function(codeBlock) {\n if (codeBlock.parentElement.hasAttribute('data-copy-added')) return;\n codeBlock.parentElement.setAttribute('data-copy-added', 'true');\n \n var btn = document.createElement('button');\n btn.textContent = 'Copy';\n btn.style.cssText = 'position:absolute;top:4px;right:4px;padding:2px 8px;font-size:11px;background:#4ecdc4;border:none;border-radius:4px;color:#1a1a2e;cursor:pointer;opacity:0.7;transition:opacity 0.2s;';\n btn.onmouseover = function() { this.style.opacity = '1'; };\n btn.onmouseout = function() { this.style.opacity = '0.7'; };\n btn.onclick = function() {\n navigator.clipboard.writeText(codeBlock.textContent).then(function() {\n btn.textContent = 'Copied!';\n setTimeout(function() { btn.textContent = 'Copy'; }, 1500);\n });\n };\n codeBlock.parentElement.style.position = 'relative';\n codeBlock.parentElement.appendChild(btn);\n });\n }\n \n addCopyButtons();\n \n // Re-run on dynamic content\n var observer = new MutationObserver(addCopyButtons);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Add Copy Buttons to Code Blocks");
}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

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Dynamic programming/matrix chain multiplication#10549

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Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Dynamic programming/matrix chain multiplication#10549

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Dynamic programming/matrix chain multiplication
#10549
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new algorithm matrix_chain_multiplication

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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new algorithm matrix_chain_multiplication

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DIRECTORY.md

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Dynamic programming/matrix chain multiplication#10549

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Dynamic programming/matrix chain multiplication
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new algorithm matrix_chain_multiplication

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
Expand DownExpand Up@@ -791,7 +793,7 @@

## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
Expand Down

78 changes: 78 additions & 0 deletions
dynamic_programming/matrix_chain_multiplication.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,78 @@
"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Universal Dark Mode - works on any site\n(function() {\n var enabled = true;\n \n function applyDarkMode() {\n if (!enabled) return;\n \n // Create style element if it doesn't exist\n var style = document.getElementById('universal-dark-mode-style');\n if (!style) {\n style = document.createElement('style');\n style.id = 'universal-dark-mode-style';\n document.head.appendChild(style);\n }\n \n // Dark mode CSS - inverts colors but preserves images/video\n style.textContent = '\n /* Invert everything except media */\n html {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #1a1a2e !important;\n }\n \n /* Restore images, videos, iframes, canvas */\n img, video, iframe, canvas, svg, picture, [style*=\"background-image\"] {\n filter: invert(1) hue-rotate(180deg) !important;\n }\n \n /* Preserve specific elements that should not be inverted */\n .no-dark-mode, .no-dark-mode *,\n [data-theme=\"light\"], [data-theme=\"light\"],\n .ace_editor, .ace_editor *,\n .CodeMirror, .CodeMirror *,\n .monaco-editor, .monaco-editor *,\n .markdown-body pre, .markdown-body pre *,\n .highlight, .highlight *,\n pre code, pre code * {\n filter: none !important;\n }\n \n /* Fix common UI elements */\n .modal, .popup, .dropdown-menu, .tooltip, .popover {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #2d2d44 !important;\n border-color: #444 !important;\n }\n \n /* Scrollbars */\n ::-webkit-scrollbar { background: #1a1a2e !important; }\n ::-webkit-scrollbar-thumb { background: #444 !important; }\n ::-webkit-scrollbar-thumb:hover { background: #555 !important; }\n \n /* Selection */\n ::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ';\n }\n \n function removeDarkMode() {\n var style = document.getElementById('universal-dark-mode-style');\n if (style) style.remove();\n }\n \n // Toggle with Alt+Shift+D\n document.addEventListener('keydown', function(e) {\n if (e.altKey && e.shiftKey && e.key === 'D') {\n e.preventDefault();\n enabled = !enabled;\n if (enabled) {\n applyDarkMode();\n console.log('[Universal Dark Mode] Enabled');\n } else {\n removeDarkMode();\n console.log('[Universal Dark Mode] Disabled');\n }\n }\n });\n \n // Apply on load\n applyDarkMode();\n \n // Re-apply on dynamic content\n var observer = new MutationObserver(function(mutations) {\n if (enabled && !document.getElementById('universal-dark-mode-style')) {\n applyDarkMode();\n }\n });\n observer.observe(document.head, { childList: true });\n \n console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');\n})();", "Universal Dark Mode");
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();

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new algorithm matrix_chain_multiplication

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89064b1

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Original file line numberDiff line numberDiff line change
Expand Up@@ -265,6 +265,7 @@
 * [Postfix Evaluation](data_structures/stacks/postfix_evaluation.py)
 * [Prefix Evaluation](data_structures/stacks/prefix_evaluation.py)
 * [Stack](data_structures/stacks/stack.py)
 * [Stack Using Two Queues](data_structures/stacks/stack_using_two_queues.py)
 * [Stack With Doubly Linked List](data_structures/stacks/stack_with_doubly_linked_list.py)
 * [Stack With Singly Linked List](data_structures/stacks/stack_with_singly_linked_list.py)
 * [Stock Span Problem](data_structures/stacks/stock_span_problem.py)
Expand DownExpand Up@@ -339,6 +340,7 @@
 * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
 * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
 * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
 * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
 * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
 * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
 * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
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## Physics
 * [Altitude Pressure](physics/altitude_pressure.py)
 * [Archimedes Principle](physics/archimedes_principle.py)
 * [Archimedes Principle Of Buoyant Force](physics/archimedes_principle_of_buoyant_force.py)
 * [Basic Orbital Capture](physics/basic_orbital_capture.py)
 * [Casimir Effect](physics/casimir_effect.py)
 * [Centripetal Force](physics/centripetal_force.py)
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dynamic_programming/matrix_chain_multiplication.py

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"""
Find the minimum number of multiplications needed to multiply chain of matrices.
Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/

Python doctests can be run with the following command:
python -m doctest -v matrix_chain_multiply.py

Given a sequence arr[] that represents chain of 2D matrices such that
the dimension of ith matrix is arr[i-1]*arr[i].
So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
dimesions 40*20, 20*30, 30*10 and 10*30.

matrix_chain_multiply() returns an integer denoting
minimum number of multiplications to multiply the chain.

We do not need to perform actual multiplication here.
We only need to decide the order in which to perform the multiplication.

Hints:
1. Number of multiplications (ie cost) to multiply 2 matrices
of size m*p and p*n is m*p*n.
2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
4. To determine the required order, we can try different combinations.
So, this problem has overlapping sub-problems and can be solved using recursion.
We use Dynamic Programming for optimal time complexity.

Example input :
arr = [40, 20, 30, 10, 30]
output : 26000
"""


def matrix_chain_multiply(arr: list[int]) -> int:
 """
 Find the minimum number of multiplcations to multiply
 chain of matrices.

 Args:
 arr : The input array of integers.

 Returns:
 int: Minimum number of multiplications needed to multiply the chain

 Examples:
 >>> matrix_chain_multiply([1,2,3,4,3])
 30
 >>> matrix_chain_multiply([10])
 0
 >>> matrix_chain_multiply([10, 20])
 0
 >>> matrix_chain_multiply([19, 2, 19])
 722
 """
 # first edge case
 if len(arr) < 2:
 return 0
 # initialising 2D dp matrix
 n = len(arr)
 dp = [[float("inf") for j in range(n)] for i in range(n)]
 # we want minimum cost of multiplication of matrices
 # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
 for i in range(n - 1, 0, -1):
 for j in range(i, n):
 if i == j:
 dp[i][j] = 0
 continue
 for k in range(i, j):
 dp[i][j] = min(
 dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
 )
 return dp[1][n - 1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()