




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
__SCREAMING_SNAKE_CASE     : Optional[Any]      ={
    """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:							dict[str, list[str]]    )       ->      set[tuple[str, str]]:


      '''simple docstring'''




      lowerCamelCase__   	=					{node: {node} for node in graph}

      lowerCamelCase__   	=					{node: graph[node][:] for node in graph}

      while len(lowerCamelCase    ) > 2:
            # Choose a random edge.
            lowerCamelCase__   	=					random.choice(list(graph_copy.keys()    )    )
            lowerCamelCase__   	=					random.choice(graph_copy[u]    )

            # Contract edge (u, v) to new node uv
            lowerCamelCase__   	=					u + v
            lowerCamelCase__   	=					list(set(graph_copy[u] + graph_copy[v]    )    )
            uv_neighbors.remove(lowerCamelCase    )
            uv_neighbors.remove(lowerCamelCase    )
            lowerCamelCase__   	=					uv_neighbors
            for neighbor in uv_neighbors:
                  graph_copy[neighbor].append(lowerCamelCase    )

            lowerCamelCase__   	=					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(lowerCamelCase    )
                  if v in graph_copy[neighbor]:
                        graph_copy[neighbor].remove(lowerCamelCase    )

    # Find cutset.
      lowerCamelCase__   	=					[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       lowerCAmelCase_      (	lowerCamelCase:							str    )       ->      int:


      '''simple docstring'''




      lowerCamelCase__   	=					str(lowerCamelCase    ).strip()
      if not bin_string:
            raise ValueError('Empty string was passed to the function'    )
      lowerCamelCase__   	=					bin_string[0] == '-'
      if is_negative:
            lowerCamelCase__   	=					bin_string[1:]
      if not all(char in '01' for char in bin_string    ):
            raise ValueError('Non-binary value was passed to the function'    )
      lowerCamelCase__   	=					0
      for char in bin_string:
            lowerCamelCase__   	=					2 * decimal_number + int(lowerCamelCase    )
      return -decimal_number if is_negative else decimal_number


if __name__ == "__main__":
      from doctest import testmod

      testmod()