


'''simple docstring'''


from __future__ import annotations

from typing import Any


def 		_lowercase   (*SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     if all(_check_not_integer(SCREAMING_SNAKE_CASE       ) for m in matrix_s       ):
          for i in matrix_s[1:]:
               _verify_matrix_sizes(matrix_s[0]					,				SCREAMING_SNAKE_CASE       )
          return [[sum(SCREAMING_SNAKE_CASE       ) for t in zip(*SCREAMING_SNAKE_CASE       )] for m in zip(*SCREAMING_SNAKE_CASE       )]
     raise TypeError("Expected a matrix, got int/list instead"       )


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     if (
         _check_not_integer(SCREAMING_SNAKE_CASE       )
         and _check_not_integer(SCREAMING_SNAKE_CASE       )
         and _verify_matrix_sizes(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       )
     ):
          return [[i - j for i, j in zip(*SCREAMING_SNAKE_CASE       )] for m in zip(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       )]
     raise TypeError("Expected a matrix, got int/list instead"       )


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     return [[x * n for x in row] for row in matrix]


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     if _check_not_integer(SCREAMING_SNAKE_CASE       ) and _check_not_integer(SCREAMING_SNAKE_CASE       ):
          __A     ,__A  :					Union[str, Any]						     =       _verify_matrix_sizes(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       )

     if cols[0] != rows[1]:
          __A  :					Optional[int]						     =       (
              "Cannot multiply matrix of dimensions "
              f"({rows[0]},{cols[0]}) and ({rows[1]},{cols[1]})"
          )
          raise ValueError(SCREAMING_SNAKE_CASE       )
     return [
         [sum(m * n for m, n in zip(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       )       ) for j in zip(*SCREAMING_SNAKE_CASE       )] for i in matrix_a
     ]


def 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     __A  :					Optional[int]						     =       int(SCREAMING_SNAKE_CASE       )
     return [[int(row == column       ) for column in range(SCREAMING_SNAKE_CASE       )] for row in range(SCREAMING_SNAKE_CASE       )]


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE = True       ):




     '''simple docstring'''





     if _check_not_integer(SCREAMING_SNAKE_CASE       ):
          if return_map:
               return map(SCREAMING_SNAKE_CASE					,				zip(*SCREAMING_SNAKE_CASE       )       )
          else:
               return list(map(SCREAMING_SNAKE_CASE					,				zip(*SCREAMING_SNAKE_CASE       )       )       )
     raise TypeError("Expected a matrix, got int/list instead"       )


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     __A  :					Dict						     =       matrix[:row] + matrix[row + 1 :]
     return [row[:column] + row[column + 1 :] for row in minor]


def 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     if len(SCREAMING_SNAKE_CASE       ) == 1:
          return matrix[0][0]

     return sum(
         x * determinant(minor(SCREAMING_SNAKE_CASE					,				0					,				SCREAMING_SNAKE_CASE       )       ) * (-1) ** i
         for i, x in enumerate(matrix[0]       )       )


def 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     __A  :					int						     =       determinant(SCREAMING_SNAKE_CASE       )
     if det == 0:
          return None

     __A  :					List[str]						     =       [
         [determinant(minor(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       )       ) for j in range(len(SCREAMING_SNAKE_CASE       )       )]
         for i in range(len(SCREAMING_SNAKE_CASE       )       )
     ]

     __A  :					Union[str, Any]						     =       [
         [x * (-1) ** (row + col) for col, x in enumerate(matrix_minor[row]       )]
         for row in range(len(SCREAMING_SNAKE_CASE       )       )
     ]
     __A  :					int						     =       list(transpose(SCREAMING_SNAKE_CASE       )       )
     return scalar_multiply(SCREAMING_SNAKE_CASE					,				1 / det       )


def 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     return not isinstance(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ) and not isinstance(matrix[0]					,				SCREAMING_SNAKE_CASE       )


def 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     return len(SCREAMING_SNAKE_CASE       ), len(matrix[0]       )


def 		_lowercase   (SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     __A  :					Union[str, Any]						     =       _shape(SCREAMING_SNAKE_CASE       ) + _shape(SCREAMING_SNAKE_CASE       )
     if shape[0] != shape[3] or shape[1] != shape[2]:
          __A  :					Tuple						     =       (
              "operands could not be broadcast together with shape "
              f"({shape[0], shape[1]}), ({shape[2], shape[3]})"
          )
          raise ValueError(SCREAMING_SNAKE_CASE       )
     return (shape[0], shape[2]), (shape[1], shape[3])


def 		_lowercase   ():




     '''simple docstring'''





     __A  :					Tuple						     =       [[12, 10], [3, 9]]
     __A  :					Dict						     =       [[3, 4], [7, 4]]
     __A  :					Optional[Any]						     =       [[11, 12, 13, 14], [21, 22, 23, 24], [31, 32, 33, 34], [41, 42, 43, 44]]
     __A  :					List[Any]						     =       [[3, 0, 2], [2, 0, -2], [0, 1, 1]]
     print(f"Add Operation, {add(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ) = } \n"       )
     print(
         f"Multiply Operation, {multiply(SCREAMING_SNAKE_CASE					,				SCREAMING_SNAKE_CASE       ) = } \n"					,				)
     print(f"Identity: {identity(5       )}\n"       )
     print(f"Minor of {matrix_c} = {minor(SCREAMING_SNAKE_CASE					,				1					,				2       )} \n"       )
     print(f"Determinant of {matrix_b} = {determinant(SCREAMING_SNAKE_CASE       )} \n"       )
     print(f"Inverse of {matrix_d} = {inverse(SCREAMING_SNAKE_CASE       )}\n"       )


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      main()





'''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          =     {
    """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 		_lowercase   (SCREAMING_SNAKE_CASE       ):




     '''simple docstring'''





     __A  :					Any						     =       {node: {node} for node in graph}

     __A  :					List[str]						     =       {node: graph[node][:] for node in graph}

     while len(SCREAMING_SNAKE_CASE       ) > 2:
          # Choose a random edge.
          __A  :					Tuple						     =       random.choice(list(graph_copy.keys()       )       )
          __A  :					Any						     =       random.choice(graph_copy[u]       )

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

          __A  :					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(SCREAMING_SNAKE_CASE       )
               if v in graph_copy[neighbor]:
                    graph_copy[neighbor].remove(SCREAMING_SNAKE_CASE       )

    # Find cutset.
     __A  :					int						     =       [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))

