





'''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
UpperCAmelCase       :		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  _a     (				lowerCAmelCase_						):





      """simple docstring"""



      _snake_case		:						Dict         =  {node: {node} for node in graph}

      _snake_case		:						List[Any]         =  {node: graph[node][:] for node in graph}

      while len(_SCREAMING_SNAKE_CASE						) > 2:
            # Choose a random edge.
            _snake_case		:						Optional[Any]         =  random.choice(list(graph_copy.keys()						)						)
            _snake_case		:						Dict         =  random.choice(graph_copy[u]						)

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

            _snake_case		:						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.
      _snake_case		:						str         =  [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))





def        _UpperCAmelCase							(	_SCREAMING_SNAKE_CASE       :		list							):





    """simple docstring"""


    if not isinstance(_SCREAMING_SNAKE_CASE			, _SCREAMING_SNAKE_CASE							):
        raise ValueError('Input series is not valid, valid series - [2, 4, 8]'							)
    if len(_SCREAMING_SNAKE_CASE							) == 0:
        raise ValueError('Input list must be a non empty list'							)
    if len(_SCREAMING_SNAKE_CASE							) == 1:
        return True
    try:
        SCREAMING_SNAKE_CASE_						=    series[1] / series[0]
        for index in range(len(_SCREAMING_SNAKE_CASE							) - 1							):
            if series[index + 1] / series[index] != common_ratio:
                return False
    except ZeroDivisionError:
        return False
    return True


def        _UpperCAmelCase							(	_SCREAMING_SNAKE_CASE       :		list							):





    """simple docstring"""


    if not isinstance(_SCREAMING_SNAKE_CASE			, _SCREAMING_SNAKE_CASE							):
        raise ValueError('Input series is not valid, valid series - [2, 4, 8]'							)
    if len(_SCREAMING_SNAKE_CASE							) == 0:
        raise ValueError('Input list must be a non empty list'							)
    SCREAMING_SNAKE_CASE_						=    1
    for value in series:
        answer *= value
    return pow(_SCREAMING_SNAKE_CASE			, 1 / len(_SCREAMING_SNAKE_CASE							)							)


if __name__ == "__main__":
       import doctest

       doctest.testmod()


