


'''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__			(						__lowerCamelCase       :							Tuple							):







   '''simple docstring'''




   _UpperCAmelCase   :   int							    ={node: {node} for node in graph}

   _UpperCAmelCase   :   str							    ={node: graph[node][:] for node in graph}

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

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

      _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(__UpperCamelCase							)
         if v in graph_copy[neighbor]:
            graph_copy[neighbor].remove(__UpperCamelCase							)

    # Find cutset.
   _UpperCAmelCase   :   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))



"""simple docstring"""


def     _lowerCamelCase(__UpperCamelCase       )				-> list:
	_lowerCAmelCase								=[True for i in range(n + 1       )]
	_lowerCAmelCase								=[i - 1 for i in range(n + 1       )]
	_lowerCAmelCase								=[]
	for i in range(2    , n + 1       ):
		if is_prime[i]:
			primes.append(__UpperCamelCase       )
		for j in range(0    , len(__UpperCamelCase       )       ):
			if i * primes[j] >= n:
				break
			_lowerCAmelCase								=False

			if i % primes[j] == 0:
				_lowerCAmelCase								=totients[i] * primes[j]
				break

			_lowerCAmelCase								=totients[i] * (primes[j] - 1)

	return totients


if __name__ == "__main__":
			import doctest

			doctest.testmod()



