





'''simple docstring'''







from __future__ import annotations

import random

# Adjacency list representation of this graph:
# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
snake_case_  :							List[str]            =	{
    "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 							lowerCamelCase_   (	SCREAMING_SNAKE_CASE__   :      dict[str, list[str]] )			->						set[tuple[str, str]]:
						UpperCAmelCase_      :   Optional[int]      =	{node: {node} for node in graph}

						UpperCAmelCase_      :   List[str]      =	{node: graph[node][:] for node in graph}

						while len(SCREAMING_SNAKE_CASE__ ) > 2:
												# Choose a random edge.
												UpperCAmelCase_      :   Optional[Any]      =	random.choice(list(graph_copy.keys() ) )
												UpperCAmelCase_      :   Optional[int]      =	random.choice(graph_copy[u] )

												# Contract edge (u, v) to new node uv
												UpperCAmelCase_      :   List[Any]      =	u + v
												UpperCAmelCase_      :   str      =	list(set(graph_copy[u] + graph_copy[v] ) )
												uv_neighbors.remove(SCREAMING_SNAKE_CASE__ )
												uv_neighbors.remove(SCREAMING_SNAKE_CASE__ )
												UpperCAmelCase_      :   Dict      =	uv_neighbors
												for neighbor in uv_neighbors:
																		graph_copy[neighbor].append(SCREAMING_SNAKE_CASE__ )

												UpperCAmelCase_      :   List[Any]      =	set(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(SCREAMING_SNAKE_CASE__ )
																		if v in graph_copy[neighbor]:
																								graph_copy[neighbor].remove(SCREAMING_SNAKE_CASE__ )

    # Find cutset.
						UpperCAmelCase_      :   int      =	[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))












'''simple docstring'''







from __future__ import annotations





def 							lowerCamelCase_   (	SCREAMING_SNAKE_CASE__   :      list[int | float] )			->						int | float:
						if len(SCREAMING_SNAKE_CASE__ ) == 0:
												raise ValueError('''find_min() arg is an empty sequence''' )
						UpperCAmelCase_      :   Optional[int]      =	nums[0]
						for num in nums:
												UpperCAmelCase_      :   Tuple      =	min(SCREAMING_SNAKE_CASE__,      SCREAMING_SNAKE_CASE__ )
						return min_num


if __name__ == "__main__":
					import doctest

					doctest.testmod(verbose=True)





