


'''simple docstring'''



def    lowerCamelCase__   ( ):




						'''simple docstring'''






						return [list(range(10_00 - i   ,     -10_00 - i   ,     -1  )  ) for i in range(10_00  )]


_lowercase    :    Optional[Any]			     =  generate_large_matrix()
_lowercase    :    Any			     =  (
    [[4, 3, 2, -1], [3, 2, 1, -1], [1, 1, -1, -2], [-1, -1, -2, -3]],
    [[3, 2], [1, 0]],
    [[7, 7, 6]],
    [[7, 7, 6], [-1, -2, -3]],
    grid,
)


def    lowerCamelCase__   ( A     :    list[list[int]]  ):




						'''simple docstring'''






						assert all(row == sorted(A   ,     reverse=A  ) for row in grid  )
						assert all(list(A  ) == sorted(A   ,     reverse=A  ) for col in zip(*A  )  )


def    lowerCamelCase__   ( A     :    list[int]  ):




						'''simple docstring'''






						UpperCAmelCase         =			0
						UpperCAmelCase         =			len(A  ) - 1

						# Edge cases such as no values or all numbers are negative.
						if not array or array[0] < 0:
												return 0

						while right + 1 > left:
												UpperCAmelCase         =			(left + right) // 2
												UpperCAmelCase         =			array[mid]

												# Num must be negative and the index must be greater than or equal to 0.
												if num < 0 and array[mid - 1] >= 0:
																		return mid

												if num >= 0:
																		UpperCAmelCase         =			mid + 1
												else:
																		UpperCAmelCase         =			mid - 1
    # No negative numbers so return the last index of the array + 1 which is the length.
						return len(A  )


def    lowerCamelCase__   ( A     :    list[list[int]]  ):




						'''simple docstring'''






						UpperCAmelCase         =			0
						UpperCAmelCase         =			len(grid[0]  )

						for i in range(len(A  )  ):
												UpperCAmelCase         =			find_negative_index(grid[i][:bound]  )
												total += bound
						return (len(A  ) * len(grid[0]  )) - total


def    lowerCamelCase__   ( A     :    list[list[int]]  ):




						'''simple docstring'''






						return len([number for row in grid for number in row if number < 0]  )


def    lowerCamelCase__   ( A     :    list[list[int]]  ):




						'''simple docstring'''






						UpperCAmelCase         =			0
						for row in grid:
												for i, number in enumerate(A  ):
																		if number < 0:
																								total += len(A  ) - i
																								break
						return total


def    lowerCamelCase__   ( ):




						'''simple docstring'''






						from timeit import timeit

						print('''Running benchmarks'''  )
						UpperCAmelCase         =			(
						    '''from __main__ import count_negatives_binary_search, '''
						    '''count_negatives_brute_force, count_negatives_brute_force_with_break, grid'''
						)
						for func in (
						    "count_negatives_binary_search",  # took 0.7727 seconds
						    "count_negatives_brute_force_with_break",  # took 4.6505 seconds
						    "count_negatives_brute_force",  # took 12.8160 seconds
						):
												UpperCAmelCase         =			timeit(f"""{func}(grid=grid)"""   ,     setup=A   ,     number=5_00  )
												print(f"""{func}() took {time:0.4f} seconds"""  )


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	benchmark()









'''simple docstring'''



import heapq


def    lowerCamelCase__   ( A     :    dict  ):




						'''simple docstring'''






						UpperCAmelCase         =			[]

						# 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(A   ,     [-1 * len(A  ), (key, value)]  )

						# chosen_vertices = set of chosen vertices
						UpperCAmelCase         =			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
												UpperCAmelCase         =			heapq.heappop(A  )[1][0]
												chosen_vertices.add(A  )

												# 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]:
																								UpperCAmelCase         =			elem[1][1].index(A  )
																								del elem[1][1][index]
																								elem[0] += 1
        # re-order the queue
												heapq.heapify(A  )
						return chosen_vertices


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	_lowercase    :    Optional[int]			     =  {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)}""")





