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priya-sundaram-dev
8e0817e829
* Clean up the networking_flow directory - Add networking_flow/README.md covering max-flow / min-cut, with a file-by-file table and guidance on which algorithm to use. - minimum_cut.py: add a module docstring with a Wikipedia URL, type hints, and corner-case doctests; work on a copy so the input graph is no longer mutated. - Add dinic.py: Dinic's algorithm (BFS level graph + DFS blocking flow), adjacency-list based so it handles parallel edges and sparse graphs. - Add push_relabel.py: the Goldberg-Tarjan push-relabel (preflow) method with highest-label selection. Both new algorithms are fully type-hinted, documented with a Wikipedia reference, and validated by doctests; their output was cross-checked against ford_fulkerson.py on thousands of random graphs. * Address review: drop __future__ import, use descriptive names, apply README wording
102 lines
3.0 KiB
Python
102 lines
3.0 KiB
Python
"""
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Minimum cut of a flow network via the Ford-Fulkerson algorithm.
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The max-flow min-cut theorem says the value of a maximum flow from the source to
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the sink equals the total capacity of the edges in a minimum s-t cut -- the
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cheapest set of edges whose removal disconnects the sink from the source. This
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module finds those cut edges: it runs Ford-Fulkerson to build the residual
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graph, then reports every original edge that goes from a vertex still reachable
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from the source to a vertex that is not.
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Reference: https://en.wikipedia.org/wiki/Minimum_cut
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See also: https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem
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"""
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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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def bfs(graph: list[list[int]], source: int, sink: int, parent: list[int]) -> bool:
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"""
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Return True if the ``sink`` is reachable from the ``source`` in the
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residual ``graph``, recording the traversal tree in ``parent``.
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>>> bfs(test_graph, 0, 5, [-1] * 6)
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True
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>>> bfs([[0, 0], [0, 0]], 0, 1, [-1, -1])
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False
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"""
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visited = [False] * len(graph)
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queue = [source]
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visited[source] = True
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while queue:
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node = queue.pop(0)
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for neighbor in range(len(graph[node])):
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if visited[neighbor] is False and graph[node][neighbor] > 0:
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queue.append(neighbor)
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visited[neighbor] = True
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parent[neighbor] = node
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return visited[sink]
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def mincut(graph: list[list[int]], source: int, sink: int) -> list[tuple[int, int]]:
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"""
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Return the edges of a minimum s-t cut as ``(from, to)`` tuples.
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The input ``graph`` is an adjacency matrix of capacities and is left
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unchanged (the algorithm works on an internal copy).
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>>> mincut(test_graph, source=0, sink=5)
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[(1, 3), (4, 3), (4, 5)]
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The capacities of the cut edges sum to the maximum flow (23 here):
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>>> sum(test_graph[u][v] for u, v in mincut(test_graph, 0, 5))
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23
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A single saturated edge is its own minimum cut:
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>>> mincut([[0, 7], [0, 0]], source=0, sink=1)
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[(0, 1)]
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"""
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residual = [row[:] for row in graph] # work on a copy; keep the input intact
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parent = [-1] * (len(residual))
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res = []
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while bfs(residual, source, sink, parent):
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path_flow = float("inf")
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s = sink
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while s != source:
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# Find the minimum residual capacity along the augmenting path.
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path_flow = min(path_flow, residual[parent[s]][s])
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s = parent[s]
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v = sink
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while v != source:
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u = parent[v]
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residual[u][v] -= path_flow
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residual[v][u] += path_flow
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v = parent[v]
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for i in range(len(graph)):
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for j in range(len(graph[0])):
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if graph[i][j] > 0 and residual[i][j] == 0:
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res.append((i, j))
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return res
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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(mincut(test_graph, source=0, sink=5))
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