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114 lines
2.9 KiB
Python
114 lines
2.9 KiB
Python
"""
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Ford-Fulkerson Algorithm for Maximum Flow Problem
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* https://en.wikipedia.org/wiki/Ford%E2%80%93Fulkerson_algorithm
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Description:
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(1) Start with initial flow as 0
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(2) Choose the augmenting path from source to sink and add the path to flow
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"""
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graph = [
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[0, 16, 13, 0, 0, 0],
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[0, 0, 10, 12, 0, 0],
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[0, 4, 0, 0, 14, 0],
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[0, 0, 9, 0, 0, 20],
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[0, 0, 0, 7, 0, 4],
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[0, 0, 0, 0, 0, 0],
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]
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def breadth_first_search(graph: list, source: int, sink: int, parents: list) -> bool:
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"""
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This function returns True if there is a node that has not iterated.
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Args:
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graph: Adjacency matrix of graph
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source: Source
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sink: Sink
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parents: Parent list
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Returns:
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True if there is a node that has not iterated.
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>>> breadth_first_search(graph, 0, 5, [-1, -1, -1, -1, -1, -1])
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True
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>>> breadth_first_search(graph, 0, 6, [-1, -1, -1, -1, -1, -1])
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Traceback (most recent call last):
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...
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IndexError: list index out of range
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"""
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visited = [False] * len(graph) # Mark all nodes as not visited
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queue = [] # breadth-first search queue
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# Source node
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queue.append(source)
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visited[source] = True
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while queue:
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u = queue.pop(0) # Pop the front node
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# Traverse all adjacent nodes of u
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for ind, node in enumerate(graph[u]):
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if visited[ind] is False and node > 0:
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queue.append(ind)
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visited[ind] = True
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parents[ind] = u
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return visited[sink]
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def ford_fulkerson(graph: list, source: int, sink: int) -> int:
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"""
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This function returns the maximum flow from source to sink in the given graph.
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CAUTION: This function changes the given graph.
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Args:
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graph: Adjacency matrix of graph
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source: Source
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sink: Sink
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Returns:
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Maximum flow
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>>> test_graph = [
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... [0, 16, 13, 0, 0, 0],
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... [0, 0, 10, 12, 0, 0],
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... [0, 4, 0, 0, 14, 0],
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... [0, 0, 9, 0, 0, 20],
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... [0, 0, 0, 7, 0, 4],
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... [0, 0, 0, 0, 0, 0],
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... ]
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>>> ford_fulkerson(test_graph, 0, 5)
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23
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"""
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# This array is filled by breadth-first search and to store path
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parent = [-1] * (len(graph))
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max_flow = 0
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# While there is a path from source to sink
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while breadth_first_search(graph, source, sink, parent):
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path_flow = int(1e9) # Infinite value
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s = sink
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while s != source:
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# Find the minimum value in the selected path
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path_flow = min(path_flow, graph[parent[s]][s])
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s = parent[s]
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max_flow += path_flow
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v = sink
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while v != source:
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u = parent[v]
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graph[u][v] -= path_flow
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graph[v][u] += path_flow
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v = parent[v]
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return max_flow
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if __name__ == "__main__":
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from doctest import testmod
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testmod()
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print(f"{ford_fulkerson(graph, source=0, sink=5) = }")
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