


'''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			:			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__      (       A   :			int     ):




						'''simple docstring'''






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

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

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

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

												UpperCAmelCase     				=	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.
						UpperCAmelCase     				=	[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




def     lowercase__			(    lowercase_						,lowercase_						,lowercase_	)    ->						int:




      """simple docstring"""

      assert n > 1 and a > 0 and greatest_common_divisor(lowercase_						,lowercase_	) == 1
      ((_UpperCamelCase),     (_UpperCamelCase),     (_UpperCamelCase))					:		int  =				extended_gcd(lowercase_						,lowercase_	)  # Implemented below
      _UpperCamelCase					:		Any  =				(b * s) % n
      return x




def     lowercase__			(    lowercase_						,lowercase_	)    ->						int:




      """simple docstring"""

      ((_UpperCamelCase),     (_UpperCamelCase))					:		List[Any]  =				extended_euclid(lowercase_						,lowercase_	)  # Implemented below
      if b < 0:
            _UpperCamelCase					:		Any  =				(b % n + n) % n
      return b




def     lowercase__			(    lowercase_						,lowercase_						,lowercase_	)    ->						int:




      """simple docstring"""

      _UpperCamelCase					:		Union[str, Any]  =				invert_modulo(lowercase_						,lowercase_	)
      _UpperCamelCase					:		Any  =				(b * s) % n
      return x




def     lowercase__			(    lowercase_						,lowercase_	)    ->						tuple[int, int, int]:




      """simple docstring"""

      assert a >= 0 and b >= 0

      if b == 0:
            _UpperCamelCase,     _UpperCamelCase,     _UpperCamelCase					:		str  =				a, 1, 0
      else:
            ((_UpperCamelCase),     (_UpperCamelCase),     (_UpperCamelCase))					:		str  =				extended_gcd(lowercase_						,a % b	)
            _UpperCamelCase					:		Dict  =				q
            _UpperCamelCase					:		Tuple  =				p - q * (a // b)

      assert a % d == 0 and b % d == 0
      assert d == a * x + b * y

      return (d, x, y)




def     lowercase__			(    lowercase_						,lowercase_	)    ->						tuple[int, int]:




      """simple docstring"""

      if b == 0:
            return (1, 0)
      ((_UpperCamelCase),     (_UpperCamelCase))					:		Optional[Any]  =				extended_euclid(lowercase_						,a % b	)
      _UpperCamelCase					:		Dict  =				a // b
      return (y, x - k * y)




def     lowercase__			(    lowercase_						,lowercase_	)    ->						int:




      """simple docstring"""

      if a < b:
            _UpperCamelCase,     _UpperCamelCase					:		List[Any]  =				b, a

      while a % b != 0:
            _UpperCamelCase,     _UpperCamelCase					:		List[Any]  =				b, a % b

      return b


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

  testmod(name="modular_division", verbose=True)
  testmod(name="modular_division2", verbose=True)
  testmod(name="invert_modulo", verbose=True)
  testmod(name="extended_gcd", verbose=True)
  testmod(name="extended_euclid", verbose=True)
  testmod(name="greatest_common_divisor", verbose=True)
