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"""

An implementation of Karger's Algorithm for partitioning a graph.

"""

import random

from typing import Dict, List, Set, Tuple

# Adjacency list representation of this graph:

# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg

TEST_GRAPH = {

"1": ["2", "3", "4", "5"],

"2": ["1", "3", "4", "5"],

"3": ["1", "2", "4", "5", "10"],

"4": ["1", "2", "3", "5", "6"],

"5": ["1", "2", "3", "4", "7"],

"6": ["7", "8", "9", "10", "4"],

"7": ["6", "8", "9", "10", "5"],

"8": ["6", "7", "9", "10"],

"9": ["6", "7", "8", "10"],

"10": ["6", "7", "8", "9", "3"],

}

def partition_graph(graph: Dict[str, List[str]]) -> Set[Tuple[str, str]]:

"""

Partitions a graph using Karger's Algorithm. Implemented from

pseudocode found here:

https://en.wikipedia.org/wiki/Karger%27s_algorithm.

This function involves random choices, meaning it will not give

consistent outputs.

Args:

graph: A dictionary containing adacency lists for the graph.

Nodes must be strings.

Returns:

The cutset of the cut found by Karger's Algorithm.

>>> graph = {'0':['1'], '1':['0']}

>>> partition_graph(graph)

{('0', '1')}

"""

# Dict that maps contracted nodes to a list of all the nodes it "contains."

contracted_nodes = {node: {node} for node in graph}

graph_copy = {node: graph[node][:] for node in graph}

while len(graph_copy) > 2:

# Choose a random edge.

u = random.choice(list(graph_copy.keys()))

v = random.choice(graph_copy[u])

# Contract edge (u, v) to new node uv

uv = u + v

uv_neighbors = list(set(graph_copy[u] + graph_copy[v]))

uv_neighbors.remove(u)

uv_neighbors.remove(v)

graph_copy[uv] = uv_neighbors

for neighbor in uv_neighbors:

graph_copy[neighbor].append(uv)

contracted_nodes[uv] = {

node for node in contracted_nodes[u].union(contracted_nodes[v])

}

# Remove nodes u and v.

del graph_copy[u]

del graph_copy[v]

for neighbor in uv_neighbors:

if u in graph_copy[neighbor]:

graph_copy[neighbor].remove(u)

if v in graph_copy[neighbor]:

graph_copy[neighbor].remove(v)

# Find cutset.

groups = [contracted_nodes[node] for node in graph_copy]

return {

(node, neighbor)

for node in groups[0]

for neighbor in graph[node]

if neighbor in groups[1]

}

if __name__ == "__main__":

print(partition_graph(TEST_GRAPH))

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