import torch
from torch import autograd





class 			__snake_case  (      autograd.Function     ):




					@staticmethod
					def       __a      (       __UpperCamelCase   ,	__UpperCamelCase   ,	__UpperCamelCase   ,	__UpperCamelCase		)					->  Optional[int]:



									'''simple docstring'''







									snake_case__ :     Dict 							=				inputs.numel()
									snake_case__ :     Tuple 							=				int(0.0_0_5 * nb_elems		) + 1
									if sigmoid:
													snake_case__ :     Dict 							=				(torch.sigmoid(__UpperCamelCase		) > threshold).type(inputs.type()		)
									else:
													snake_case__ :     List[Any] 							=				(inputs > threshold).type(inputs.type()		)
									if mask.sum() < nb_min:
													# We limit the pruning so that at least 0.5% (half a percent) of the weights are remaining
													snake_case__ :     int 							=				inputs.flatten().kthvalue(max(nb_elems - nb_min   ,	1		)		).values
													snake_case__ :     List[str] 							=				(inputs > k_threshold).type(inputs.type()		)
									return mask




					@staticmethod
					def       __a      (       __UpperCamelCase   ,	__UpperCamelCase		)					->  List[str]:



									'''simple docstring'''







									return gradOutput, None, None





class 			__snake_case  (      autograd.Function     ):




					@staticmethod
					def       __a      (       __UpperCamelCase   ,	__UpperCamelCase   ,	__UpperCamelCase		)					->  str:



									'''simple docstring'''







									snake_case__ :     str 							=				inputs.clone()
									snake_case__							,  snake_case__ :     List[str] 							=				inputs.flatten().sort(descending=__UpperCamelCase		)
									snake_case__ :     List[Any] 							=				int(threshold * inputs.numel()		)

									# flat_out and mask access the same memory.
									snake_case__ :     Union[str, Any] 							=				mask.flatten()
									snake_case__ :     List[str] 							=				0
									snake_case__ :     Optional[int] 							=				1
									return mask




					@staticmethod
					def       __a      (       __UpperCamelCase   ,	__UpperCamelCase		)					->  List[str]:



									'''simple docstring'''







									return gradOutput, None





class 			__snake_case  (      _lowerCamelCase     ):




					@staticmethod
					def       __a      (       __UpperCamelCase   ,	__UpperCamelCase		)					->  Dict:



									'''simple docstring'''







									snake_case__ :     Any 							=				inputs.clone()
									snake_case__							,  snake_case__ :     Optional[Any] 							=				inputs.abs().flatten().sort(descending=__UpperCamelCase		)
									snake_case__ :     Tuple 							=				int(threshold * inputs.numel()		)

									# flat_out and mask access the same memory.
									snake_case__ :     Optional[int] 							=				mask.flatten()
									snake_case__ :     Dict 							=				0
									snake_case__ :     Any 							=				1
									return mask



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
lowerCAmelCase__					:       Dict		=       {
    '''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 			UpperCamelCase__						(    A__   )       -> set[tuple[str, str]]:
				snake_case__ :     Tuple 							=				{node: {node} for node in graph}

				snake_case__ :     Tuple 							=				{node: graph[node][:] for node in graph}

				while len(A__   ) > 2:
								# Choose a random edge.
								snake_case__ :     Union[str, Any] 							=				random.choice(list(graph_copy.keys()   )   )
								snake_case__ :     Any 							=				random.choice(graph_copy[u]   )

								# Contract edge (u, v) to new node uv
								snake_case__ :     Tuple 							=				u + v
								snake_case__ :     Dict 							=				list(set(graph_copy[u] + graph_copy[v]   )   )
								uv_neighbors.remove(A__   )
								uv_neighbors.remove(A__   )
								snake_case__ :     Optional[Any] 							=				uv_neighbors
								for neighbor in uv_neighbors:
												graph_copy[neighbor].append(A__   )

								snake_case__ :     Tuple 							=				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(A__   )
												if v in graph_copy[neighbor]:
																graph_copy[neighbor].remove(A__   )

    # Find cutset.
				snake_case__ :     Tuple 							=				[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))


