

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 _a			(	SCREAMING_SNAKE_CASE		:      Optional[int]		)   ->   set[tuple[str, str]]:

 """simple docstring"""






 __lowerCAmelCase:       Optional[Any]            =				{node: {node} for node in graph}

 __lowerCAmelCase:       Optional[Any]            =				{node: graph[node][:] for node in graph}

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

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

  __lowerCAmelCase:       List[str]            =				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(lowercase_		)
   if v in graph_copy[neighbor]:
    graph_copy[neighbor].remove(lowercase_		)

    # Find cutset.
 __lowerCAmelCase:       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))








from __future__ import annotations

from collections.abc import Iterable
from dataclasses import dataclass
from typing import Any






class 							A_			:
     _lowercase      :					Any       =     ('_front', '_back', '_len')






     @dataclass
     class 							A_			:
          _lowercase      :					Any       =     None
          _lowercase      :					Deque._Node | None       =     None
          _lowercase      :					Deque._Node | None       =     None







     class 							A_			:
          _lowercase      :					Optional[Any]       =     ('_cur',)


          def __init__(							self      :		Any      ,  UpperCAmelCase      :		Deque._Node | None )							->     None:
           __lowerCAmelCase:       int            =				cur


          def __iter__(							self      :		int )							->     Deque._Iterator:
           return self


          def   UpperCAmelCase  (							self      :		Optional[Any] )							->     Any:
           if self._cur is None:
            # finished iterating
            raise StopIteration
           __lowerCAmelCase:       Optional[Any]            =				self._cur.val
           __lowerCAmelCase:       List[Any]            =				self._cur.next_node

           return val







     def __init__(							self      :		Any      ,  UpperCAmelCase      :		Iterable[Any] | None = None )							->     None:
      __lowerCAmelCase:       Any            =				None
      __lowerCAmelCase:       Any            =				None
      __lowerCAmelCase:       int            =				0

      if iterable is not None:
       # append every value to the deque
       for val in iterable:
        self.append(UpperCAmelCase )


     def   UpperCAmelCase  (							self      :		List[str]      ,  UpperCAmelCase      :		Any )							->     None:
      __lowerCAmelCase:       Union[str, Any]            =				self._Node(UpperCAmelCase      ,  UpperCAmelCase      ,  UpperCAmelCase )
      if self.is_empty():
       # front = back
       __lowerCAmelCase:       int            =				node
       __lowerCAmelCase:       List[Any]            =				1
      else:
       # connect nodes
       __lowerCAmelCase:       Optional[Any]            =				node
       __lowerCAmelCase:       str            =				self._back
       __lowerCAmelCase:       Optional[int]            =				node  # assign new back to the new node

       self._len += 1

       # make sure there were no errors
       assert not self.is_empty(), "Error on appending value."


     def   UpperCAmelCase  (							self      :		Optional[int]      ,  UpperCAmelCase      :		Any )							->     None:
      __lowerCAmelCase:       int            =				self._Node(UpperCAmelCase      ,  UpperCAmelCase      ,  UpperCAmelCase )
      if self.is_empty():
       # front = back
       __lowerCAmelCase:       Union[str, Any]            =				node
       __lowerCAmelCase:       str            =				1
      else:
       # connect nodes
       __lowerCAmelCase:       Dict            =				self._front
       __lowerCAmelCase:       Optional[int]            =				node
       __lowerCAmelCase:       Any            =				node  # assign new front to the new node

       self._len += 1

       # make sure there were no errors
       assert not self.is_empty(), "Error on appending value."


     def   UpperCAmelCase  (							self      :		Union[str, Any]      ,  UpperCAmelCase      :		Iterable[Any] )							->     None:
      for val in iterable:
       self.append(UpperCAmelCase )


     def   UpperCAmelCase  (							self      :		List[str]      ,  UpperCAmelCase      :		Iterable[Any] )							->     None:
      for val in iterable:
       self.appendleft(UpperCAmelCase )


     def   UpperCAmelCase  (							self      :		Dict )							->     Any:
      assert not self.is_empty(), "Deque is empty."

      __lowerCAmelCase:       Optional[int]            =				self._back
      __lowerCAmelCase:       Optional[Any]            =				self._back.prev_node  # set new back
      # drop the last node - python will deallocate memory automatically
      __lowerCAmelCase:       Optional[int]            =				None

      self._len -= 1

      return topop.val


     def   UpperCAmelCase  (							self      :		Dict )							->     Any:
      assert not self.is_empty(), "Deque is empty."

      __lowerCAmelCase:       Optional[Any]            =				self._front
      __lowerCAmelCase:       Optional[Any]            =				self._front.next_node  # set new front and drop the first node
      __lowerCAmelCase:       int            =				None

      self._len -= 1

      return topop.val


     def   UpperCAmelCase  (							self      :		Dict )							->     bool:
      return self._front is None


     def __len__(							self      :		List[Any] )							->     int:
      return self._len


     def __eq__(							self      :		Union[str, Any]      ,  UpperCAmelCase      :		object )							->     bool:

      if not isinstance(UpperCAmelCase      ,  UpperCAmelCase ):
       return NotImplemented

      __lowerCAmelCase:       List[Any]            =				self._front
      __lowerCAmelCase:       Dict            =				other._front

      # if the length of the dequeues are not the same, they are not equal
      if len(self ) != len(UpperCAmelCase ):
       return False

      while me is not None and oth is not None:
       # compare every value
       if me.val != oth.val:
        return False
       __lowerCAmelCase:       Optional[Any]            =				me.next_node
       __lowerCAmelCase:       Optional[Any]            =				oth.next_node

      return True


     def __iter__(							self      :		Tuple )							->     Deque._Iterator:
      return Deque._Iterator(self._front )


     def __repr__(							self      :		Optional[Any] )							->     str:
      __lowerCAmelCase:       Union[str, Any]            =				[]
      __lowerCAmelCase:       Optional[int]            =				self._front
      while aux is not None:
       # append the values in a list to display
       values_list.append(aux.val )
       __lowerCAmelCase:       Any            =				aux.next_node

      return F'''[{', '.join(repr(UpperCAmelCase ) for val in values_list )}]'''


if __name__ == "__main__":
  import doctest

  doctest.testmod()





