






def     _a					(       UpperCamelCase_   :			list[list[int]]      ,				UpperCamelCase_   :			list[list[int]]				)	->				list[list[int]]:
						"""simple docstring"""







						lowerCAmelCase__	 =  []
						lowerCAmelCase__	 =  len(_lowerCamelCase				)
						for i in range(_lowerCamelCase				):
												lowerCAmelCase__	 =  []
												for j in range(_lowerCamelCase				):
																		lowerCAmelCase__	 =  0
																		for k in range(_lowerCamelCase				):
																								lowerCAmelCase__	 =  val + matrix_a[i][k] * matrix_b[k][j]
																		list_a.append(_lowerCamelCase				)
												matrix_c.append(_lowerCamelCase				)
						return matrix_c


def     _a					(       UpperCamelCase_   :			int				)	->				list[list[int]]:
						"""simple docstring"""







						return [[int(row == column				) for column in range(_lowerCamelCase				)] for row in range(_lowerCamelCase				)]


def     _a					(       UpperCamelCase_   :			int				)	->				int:
						"""simple docstring"""







						if n <= 1:
												return n
						lowerCAmelCase__	 =  identity(2				)
						lowerCAmelCase__	 =  [[1, 1], [1, 0]]
						lowerCAmelCase__	 =  n - 1
						while n > 0:
												if n % 2 == 1:
																		lowerCAmelCase__	 =  multiply(_lowerCamelCase      ,				_lowerCamelCase				)
												lowerCAmelCase__	 =  multiply(_lowerCamelCase      ,				_lowerCamelCase				)
												lowerCAmelCase__	 =  int(n / 2				)
						return res_matrix[0][0]


def     _a					(       UpperCamelCase_   :			int				)	->				int:
						"""simple docstring"""







						if n <= 1:
												return n
						lowerCAmelCase__	 =  0
						lowerCAmelCase__	 =  1
						for _ in range(2      ,				n + 1				):
												lowerCAmelCase__ ,							lowerCAmelCase__	 =  fiba, fiba + fiba
						return fiba


def     _a					(       )	->				None:
						"""simple docstring"""







						for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
												lowerCAmelCase__	 =  int("".join(c for c in ordinal if c in "0123456789"				)				)  # 1000th --> 1000
												print(
												    F"{ordinal} fibonacci number using matrix exponentiation is "
												    F"{nth_fibonacci_matrix(_lowerCamelCase				)} and using bruteforce is "
												    F"{nth_fibonacci_bruteforce(_lowerCamelCase				)}\n"				)
						# from timeit import timeit
						# print(timeit("nth_fibonacci_matrix(1000000)",
						#              "from main import nth_fibonacci_matrix", number=5))
						# print(timeit("nth_fibonacci_bruteforce(1000000)",
						#              "from main import nth_fibonacci_bruteforce", number=5))
						# 2.3342058970001744
						# 57.256506615000035


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				main()














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					(       UpperCamelCase_   :			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(UpperCamelCase_				) > 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(UpperCamelCase_				)
												uv_neighbors.remove(UpperCamelCase_				)
												lowerCAmelCase__	 =  uv_neighbors
												for neighbor in uv_neighbors:
																		graph_copy[neighbor].append(UpperCamelCase_				)

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

    # 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))






