





'''simple docstring'''



from __future__ import annotations

import math




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list						,    lowerCamelCase_     :					list			):
			if len(lowerCamelCase_			) != 2 or len(a[0]			) != 2 or len(lowerCamelCase_			) != 2 or len(b[0]			) != 2:
						raise Exception('''Matrices are not 2x2'''			)
			__lowercase			  =							[
			    [a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
			    [a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]],
			]
			return new_matrix




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list						,    lowerCamelCase_     :					list			):
			return [
			    [matrix_a[row][col] + matrix_b[row][col] for col in range(len(matrix_a[row]			)			)]
			    for row in range(len(lowerCamelCase_			)			)
			]




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list						,    lowerCamelCase_     :					list			):
			return [
			    [matrix_a[row][col] - matrix_b[row][col] for col in range(len(matrix_a[row]			)			)]
			    for row in range(len(lowerCamelCase_			)			)
			]




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list			):
			if len(lowerCamelCase_			) % 2 != 0 or len(a[0]			) % 2 != 0:
						raise Exception('''Odd matrices are not supported!'''			)

			__lowercase			  =							len(lowerCamelCase_			)
			__lowercase			  =							matrix_length // 2

			__lowercase			  =							[[a[i][j] for j in range(lowerCamelCase_						,    lowerCamelCase_			)] for i in range(lowerCamelCase_			)]
			__lowercase			  =							[
			    [a[i][j] for j in range(lowerCamelCase_						,    lowerCamelCase_			)] for i in range(lowerCamelCase_						,    lowerCamelCase_			)
			]

			__lowercase			  =							[[a[i][j] for j in range(lowerCamelCase_			)] for i in range(lowerCamelCase_			)]
			__lowercase			  =							[[a[i][j] for j in range(lowerCamelCase_			)] for i in range(lowerCamelCase_						,    lowerCamelCase_			)]

			return top_left, top_right, bot_left, bot_right




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list			):
			return len(lowerCamelCase_			), len(matrix[0]			)




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list			):
			print('''\n'''.join(str(lowerCamelCase_			) for line in matrix			)			)




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list						,    lowerCamelCase_     :					list			):
			if matrix_dimensions(lowerCamelCase_			) == (2, 2):
						return default_matrix_multiplication(lowerCamelCase_						,    lowerCamelCase_			)

			__lowercase  ,						__lowercase  ,						__lowercase  ,						__lowercase			  =							split_matrix(lowerCamelCase_			)
			__lowercase  ,						__lowercase  ,						__lowercase  ,						__lowercase			  =							split_matrix(lowerCamelCase_			)

			__lowercase			  =							actual_strassen(lowerCamelCase_						,    matrix_subtraction(lowerCamelCase_						,    lowerCamelCase_			)			)
			__lowercase			  =							actual_strassen(matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)						,    lowerCamelCase_			)
			__lowercase			  =							actual_strassen(matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)						,    lowerCamelCase_			)
			__lowercase			  =							actual_strassen(lowerCamelCase_						,    matrix_subtraction(lowerCamelCase_						,    lowerCamelCase_			)			)
			__lowercase			  =							actual_strassen(matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)						,    matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)			)
			__lowercase			  =							actual_strassen(matrix_subtraction(lowerCamelCase_						,    lowerCamelCase_			)						,    matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)			)
			__lowercase			  =							actual_strassen(matrix_subtraction(lowerCamelCase_						,    lowerCamelCase_			)						,    matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)			)

			__lowercase			  =							matrix_addition(matrix_subtraction(matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)						,    lowerCamelCase_			)						,    lowerCamelCase_			)
			__lowercase			  =							matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)
			__lowercase			  =							matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)
			__lowercase			  =							matrix_subtraction(matrix_subtraction(matrix_addition(lowerCamelCase_						,    lowerCamelCase_			)						,    lowerCamelCase_			)						,    lowerCamelCase_			)

			# construct the new matrix from our 4 quadrants
			__lowercase			  =							[]
			for i in range(len(lowerCamelCase_			)			):
						new_matrix.append(top_left[i] + top_right[i]			)
			for i in range(len(lowerCamelCase_			)			):
						new_matrix.append(bot_left[i] + bot_right[i]			)
			return new_matrix




def 				_lowerCAmelCase  (			lowerCamelCase_     :					list						,    lowerCamelCase_     :					list			):
			if matrix_dimensions(lowerCamelCase_			)[1] != matrix_dimensions(lowerCamelCase_			)[0]:
						__lowercase			  =							(
						    '''Unable to multiply these matrices, please check the dimensions.\n'''
						    f"Matrix A: {matrixa}\n"
						    f"Matrix B: {matrixa}"
						)
						raise Exception(lowerCamelCase_			)
			__lowercase			  =							matrix_dimensions(lowerCamelCase_			)
			__lowercase			  =							matrix_dimensions(lowerCamelCase_			)

			if dimensiona[0] == dimensiona[1] and dimensiona[0] == dimensiona[1]:
						return [matrixa, matrixa]

			__lowercase			  =							max(*lowerCamelCase_						,    *lowerCamelCase_			)
			__lowercase			  =							int(math.pow(2						,    math.ceil(math.loga(lowerCamelCase_			)			)			)			)
			__lowercase			  =							matrixa
			__lowercase			  =							matrixa

			# Adding zeros to the matrices so that the arrays dimensions are the same and also
			# power of 2
			for i in range(0						,    lowerCamelCase_			):
						if i < dimensiona[0]:
									for _ in range(dimensiona[1]						,    lowerCamelCase_			):
												new_matrixa[i].append(0			)
						else:
									new_matrixa.append([0] * maxim			)
						if i < dimensiona[0]:
									for _ in range(dimensiona[1]						,    lowerCamelCase_			):
												new_matrixa[i].append(0			)
						else:
									new_matrixa.append([0] * maxim			)

			__lowercase			  =							actual_strassen(lowerCamelCase_						,    lowerCamelCase_			)

			# Removing the additional zeros
			for i in range(0						,    lowerCamelCase_			):
						if i < dimensiona[0]:
									for _ in range(dimensiona[1]						,    lowerCamelCase_			):
												final_matrix[i].pop()
						else:
									final_matrix.pop()
			return final_matrix


if __name__ == "__main__":
	_SCREAMING_SNAKE_CASE   =      [
	    [2, 3, 4, 5],
	    [6, 4, 3, 1],
	    [2, 3, 6, 7],
	    [3, 1, 2, 4],
	    [2, 3, 4, 5],
	    [6, 4, 3, 1],
	    [2, 3, 6, 7],
	    [3, 1, 2, 4],
	    [2, 3, 4, 5],
	    [6, 2, 3, 1],
	]
	_SCREAMING_SNAKE_CASE   =      [[0, 2, 1, 1], [1_6, 2, 3, 3], [2, 2, 7, 7], [1_3, 1_1, 2_2, 4]]
	print(strassen(matrixa, matrixa))












'''simple docstring'''



import heapq




def 				_lowerCAmelCase  (			lowerCamelCase_     :					dict			):
			__lowercase			  =							[]

			# for each node and his adjacency list add them and the rank of the node to queue
			# using heapq module the queue will be filled like a Priority Queue
			# heapq works with a min priority queue, so I used -1*len(v) to build it
			for key, value in graph.items():
						# O(log(n))
						heapq.heappush(lowerCamelCase_						,    [-1 * len(lowerCamelCase_			), (key, value)]			)

			# chosen_vertices = set of chosen vertices
			__lowercase			  =							set()

			# while queue isn't empty and there are still edges
			#   (queue[0][0] is the rank of the node with max rank)
			while queue and queue[0][0] != 0:
						# extract vertex with max rank from queue and add it to chosen_vertices
						__lowercase			  =							heapq.heappop(lowerCamelCase_			)[1][0]
						chosen_vertices.add(lowerCamelCase_			)

						# Remove all arcs adjacent to argmax
						for elem in queue:
									# if v haven't adjacent node, skip
									if elem[0] == 0:
												continue
									# if argmax is reachable from elem
									# remove argmax from elem's adjacent list and update his rank
									if argmax in elem[1][1]:
												__lowercase			  =							elem[1][1].index(lowerCamelCase_			)
												del elem[1][1][index]
												elem[0] += 1
        # re-order the queue
						heapq.heapify(lowerCamelCase_			)
			return chosen_vertices


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	_SCREAMING_SNAKE_CASE   =      {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
	print(f'''Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}''')





