"""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
__A         =				{
    "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 SCREAMING_SNAKE_CASE__    (     __UpperCAmelCase				)						->						set[tuple[str, str]]:
   lowercase__:     Union[str, Any]      =   {node: {node} for node in graph}

   lowercase__:     Dict      =   {node: graph[node][:] for node in graph}

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

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

      lowercase__:     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.
   lowercase__:     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))

"""simple docstring"""






from __future__ import annotations

import random
from collections.abc import Iterable
from typing import Any, Generic, TypeVar

__A         =				TypeVar("T", bound=bool)


class 					UpperCAmelCase		(Generic[T]						):



    """simple docstring"""

    def __init__(     self					,    _UpperCAmelCase							):
       lowercase__:     T      =   value
       lowercase__:     RandomizedHeapNode[T] | None      =   None
       lowercase__:     RandomizedHeapNode[T] | None      =   None






    @property
    def _snake_case    (     self							):
       return self._value






    @staticmethod
    def _snake_case    (     _UpperCAmelCase					,    _UpperCAmelCase							):
       if not roota:
          return roota

       if not roota:
          return roota

       if roota.value > roota.value:
          lowercase__,				lowercase__:     List[str]      =   roota, roota

       if random.choice([True, False]							):
          lowercase__,				lowercase__:     Any      =   roota.right, roota.left

       lowercase__:     Optional[int]      =   RandomizedHeapNode.merge(roota.left					,    _UpperCAmelCase							)

       return roota



class 					UpperCAmelCase		(Generic[T]						):



    """simple docstring"""

    def __init__(     self					,    _UpperCAmelCase = ()							):
       lowercase__:     RandomizedHeapNode[T] | None      =   None

       if data:
          for item in data:
             self.insert(_UpperCAmelCase							)






    def _snake_case    (     self					,    _UpperCAmelCase							):
       lowercase__:     Any      =   RandomizedHeapNode.merge(self._root					,    RandomizedHeapNode(_UpperCAmelCase							)							)






    def _snake_case    (     self							):
       lowercase__:     Union[str, Any]      =   self.top()

       if self._root is None:
          return None

       lowercase__:     int      =   RandomizedHeapNode.merge(self._root.left					,    self._root.right							)

       return result






    def _snake_case    (     self							):
       if not self._root:
          raise IndexError('''Can\'t get top element for the empty heap.'''							)
       return self._root.value






    def _snake_case    (     self							):
       lowercase__:     Dict      =   None






    def _snake_case    (     self							):
       lowercase__:     Optional[int]      =   []
       while self:
          result.append(self.pop()							)

       return result






    def __bool__(     self							):
       return self._root is not None


if __name__ == "__main__":
       import doctest

       doctest.testmod()
