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Added vertex-limited network flow reduction #8720

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Added vertex-limited network flow reduction
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48 changes: 48 additions & 0 deletions networking_flow/vertex_limited.py
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from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):

@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited

Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:

Please provide type hint for the parameter: graph

Please provide type hint for the parameter: limits

max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

# Give all directed edges infinite capacity
for src_vertex in range(len(graph)):
for dest_vertex in range(len(graph[0])):
if graph[src_vertex][dest_vertex] == 1:
max_flow_graph[src_vertex][dest_vertex] = float("inf")

# Expand all vertices into two vertices
for src_vertex in range(len(graph)):
# move all outbound edges of the vertex to the new vertex
max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

# create edge with capacity equal to the vertex's limit
# from the original vertex to the new vertex
max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
[0, 1, 1, 0, 0, 0],
[0, 0, 1, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 1, 0, 1],
[0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19
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Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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Added vertex-limited network flow reduction

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Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

May 10, 2023
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Module import fix

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Force GitHub README to respect dark mode
(function() {
var style = document.createElement('style');
style.textContent = '
.markdown-body {
color-scheme: dark light;
}
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Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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Added vertex-limited network flow reduction#8720

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Added vertex-limited network flow reduction
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b6d28fd

Added vertex-limited network flow reduction

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51833ce

Added wikipedia link for explanation

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pre-commit-ci[bot] May 10, 2023
b80ad47

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f6eae02

Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

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99565cb

Module import fix

May 10, 2023

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
(function() {
var ref = document.referrer;
var terms = [];
if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {
var url = new URL(ref);
var q = url.searchParams.get('q') || url.searchParams.get('p');
if (q) {
terms = q.split(/\s+/).filter(function(t) { return t.length > 2; });
}
}
if (terms.length === 0) return;
var style = document.createElement('style');
style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';
document.head.appendChild(style);
function highlight(node) {
if (node.nodeType === 3) { // text node
var text = node.textContent;
var found = false;
terms.forEach(function(term) {
var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\]\\]/g, '\\') + ')', 'gi');
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span.className = 'userscript-highlight';
span.textContent = part;
frag.appendChild(span);
}
});
node.parentNode.replaceChild(frag, node);
}
});
} else if (node.nodeType === 1 && node.childNodes) { // element
var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];
if (!skipTags.includes(node.tagName)) {
Array.from(node.childNodes).forEach(highlight);
}
}
}
highlight(document.body);
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mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
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observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
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var __re = new RegExp('^' + ".*" + '
Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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#8720
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b6d28fd

Added vertex-limited network flow reduction

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51833ce

Added wikipedia link for explanation

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2c2e53c

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pre-commit-ci[bot] May 10, 2023
b80ad47

Pre-commit fix

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Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
(function() {
var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',
'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',
'ref', 'ref_src', 'source', 'medium', 'campaign'];
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try {
var u = new URL(url, window.location.origin);
var changed = false;
trackingParams.forEach(function(p) {
if (u.searchParams.has(p)) {
u.searchParams.delete(p);
changed = true;
}
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} catch (e) {
return url;
}
}
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var clean = cleanUrl(a.href);
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});
}
cleanLinks();
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mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1) {
if (node.tagName === 'A') cleanLinks();
node.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '
Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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b6d28fd

Added vertex-limited network flow reduction

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51833ce

Added wikipedia link for explanation

May 10, 2023
2c2e53c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] May 10, 2023
b80ad47

Pre-commit fix

May 10, 2023
f6eae02

Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

May 10, 2023
99565cb

Module import fix

May 10, 2023

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
(function() {
function tryTheater() {
var btn = document.querySelector('button[aria-label="Theater mode"], ytd-player #player button[title="Theater mode"]');
if (btn && !btn.classList.contains('activated')) {
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Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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Added vertex-limited network flow reduction

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networking_flow/vertex_limited.py

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from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

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Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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Added vertex-limited network flow reduction#8720

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Added vertex-limited network flow reduction
#8720
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b6d28fd

Added vertex-limited network flow reduction

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51833ce

Added wikipedia link for explanation

May 10, 2023
2c2e53c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] May 10, 2023
b80ad47

Pre-commit fix

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f6eae02

Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

May 10, 2023
99565cb

Module import fix

May 10, 2023

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19

, 'i');
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/* Invert everything except media */
html {
filter: invert(1) hue-rotate(180deg) !important;
background: #1a1a2e !important;
}
/* Restore images, videos, iframes, canvas */
img, video, iframe, canvas, svg, picture, [style*="background-image"] {
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';
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}
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applyDarkMode();
}
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console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');
})();
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
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Added vertex-limited network flow reduction by iiloni-ce · Pull Request #8720 · TheAlgorithms/Python · GitHub
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Added vertex-limited network flow reduction#8720

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Added vertex-limited network flow reduction
#8720
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b6d28fd

Added vertex-limited network flow reduction

May 10, 2023
51833ce

Added wikipedia link for explanation

May 10, 2023
2c2e53c

[pre-commit.ci] auto fixes from pre-commit.com hooks


pre-commit-ci[bot] May 10, 2023
b80ad47

Pre-commit fix

May 10, 2023
f6eae02

Merge branch 'vertex-limited' of github.com:iiloni-umd/Python into ve…

May 10, 2023
99565cb

Module import fix

May 10, 2023

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48 changes: 48 additions & 0 deletions
networking_flow/vertex_limited.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,48 @@
from networking_flow.ford_fulkerson import ford_fulkerson

# Max-Flow reduction for vertex-limited networks (rather than edge-limited)
"""
Reduces a network that has vertex capacities into a network with edge capacities
so that Ford-Fulkerson will work.
Explanation: https://en.wikipedia.org/wiki/Maximum_flow_problem#Maximum_flow_with_vertex_capacities
"""


def vertex_limited(graph, limits):
@algorithms-keeperalgorithms-keeperBotMay 10, 2023

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As there is no test file in this pull request nor any test function or class in the file networking_flow/vertex_limited.py, please provide doctest for the function vertex_limited
Please provide return type hint for the function: vertex_limited. If the function does not return a value, please provide the type hint as:def function() -> None:
Please provide type hint for the parameter: graph
Please provide type hint for the parameter: limits
 max_flow_graph = [[0 for i in range(2 * len(graph))] for j in range(2 * len(graph))]

 # Give all directed edges infinite capacity
 for src_vertex in range(len(graph)):
 for dest_vertex in range(len(graph[0])):
 if graph[src_vertex][dest_vertex] == 1:
 max_flow_graph[src_vertex][dest_vertex] = float("inf")

 # Expand all vertices into two vertices
 for src_vertex in range(len(graph)):
 # move all outbound edges of the vertex to the new vertex
 max_flow_graph[src_vertex + len(graph)] = max_flow_graph[src_vertex]
 max_flow_graph[src_vertex] = [0 for i in range(len(max_flow_graph))]

 # create edge with capacity equal to the vertex's limit
 # from the original vertex to the new vertex
 max_flow_graph[src_vertex][src_vertex + len(graph)] = limits[src_vertex]

 return max_flow_graph


# Test Case


limits = [29, 12, 14, 20, 7, 29]
graph = [ # rows are source, columns are destination
 [0, 1, 1, 0, 0, 0],
 [0, 0, 1, 1, 0, 0],
 [0, 1, 0, 0, 1, 0],
 [0, 0, 1, 0, 0, 1],
 [0, 0, 0, 1, 0, 1],
 [0, 0, 0, 0, 0, 0],
]

mf_graph = vertex_limited(graph, limits)
print(mf_graph) # prints graph reduced to a standard flow network
print(ford_fulkerson(mf_graph, 0, len(mf_graph) - 1)) # max flow should be 19