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88 lines
2.6 KiB
Python
88 lines
2.6 KiB
Python
"""
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An implementation of Karger's Algorithm for partitioning a graph.
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"""
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import random
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from typing import Dict, List, Set, Tuple
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# Adjacency list representation of this graph:
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# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
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TEST_GRAPH = {
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"1": ["2", "3", "4", "5"],
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"2": ["1", "3", "4", "5"],
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"3": ["1", "2", "4", "5", "10"],
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"4": ["1", "2", "3", "5", "6"],
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"5": ["1", "2", "3", "4", "7"],
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"6": ["7", "8", "9", "10", "4"],
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"7": ["6", "8", "9", "10", "5"],
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"8": ["6", "7", "9", "10"],
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"9": ["6", "7", "8", "10"],
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"10": ["6", "7", "8", "9", "3"],
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}
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def partition_graph(graph: Dict[str, List[str]]) -> Set[Tuple[str, str]]:
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"""
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Partitions a graph using Karger's Algorithm. Implemented from
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pseudocode found here:
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https://en.wikipedia.org/wiki/Karger%27s_algorithm.
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This function involves random choices, meaning it will not give
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consistent outputs.
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Args:
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graph: A dictionary containing adacency lists for the graph.
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Nodes must be strings.
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Returns:
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The cutset of the cut found by Karger's Algorithm.
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>>> graph = {'0':['1'], '1':['0']}
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>>> partition_graph(graph)
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{('0', '1')}
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"""
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# Dict that maps contracted nodes to a list of all the nodes it "contains."
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contracted_nodes = {node: {node} for node in graph}
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graph_copy = {node: graph[node][:] for node in graph}
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while len(graph_copy) > 2:
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# Choose a random edge.
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u = random.choice(list(graph_copy.keys()))
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v = random.choice(graph_copy[u])
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# Contract edge (u, v) to new node uv
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uv = u + v
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uv_neighbors = list(set(graph_copy[u] + graph_copy[v]))
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uv_neighbors.remove(u)
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uv_neighbors.remove(v)
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graph_copy[uv] = uv_neighbors
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for neighbor in uv_neighbors:
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graph_copy[neighbor].append(uv)
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contracted_nodes[uv] = {
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node for node in contracted_nodes[u].union(contracted_nodes[v])
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}
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# Remove nodes u and v.
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del graph_copy[u]
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del graph_copy[v]
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for neighbor in uv_neighbors:
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if u in graph_copy[neighbor]:
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graph_copy[neighbor].remove(u)
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if v in graph_copy[neighbor]:
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graph_copy[neighbor].remove(v)
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# Find cutset.
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groups = [contracted_nodes[node] for node in graph_copy]
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return {
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(node, neighbor)
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for node in groups[0]
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for neighbor in graph[node]
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if neighbor in groups[1]
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}
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if __name__ == "__main__":
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print(partition_graph(TEST_GRAPH))
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