






"""simple docstring"""

def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase )		->					Optional[int]:

		"""simple docstring"""







		lowerCAmelCase_     :							List[str]             =    1
		while b > 0:
				if b & 1:
						res *= a

				a *= a
				b >>= 1

		return res

def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )		->					List[str]:

		"""simple docstring"""







		lowerCAmelCase_     :							str             =    1
		while b > 0:
				if b & 1:
						lowerCAmelCase_     :							Optional[Any]             =    ((res % c) * (a % c)) % c

				a *= a
				b >>= 1

		return res








"""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
lowercase__      =				{
    """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			(						__UpperCamelCase )		->					set[tuple[str, str]]:

		"""simple docstring"""







		lowerCAmelCase_     :							Union[str, Any]             =    {node: {node} for node in graph}

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

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

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

				lowerCAmelCase_     :							Optional[int]             =    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(__UpperCamelCase )
						if v in graph_copy[neighbor]:
								graph_copy[neighbor].remove(__UpperCamelCase )

    # Find cutset.
		lowerCAmelCase_     :							Union[str, Any]             =    [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))
