def A__  (			)				->		list[list[int]]:
       return [list(range(1_000 - i,		-1_000 - i,		-1					)					) for i in range(1_000					)]


_lowercase							:      str      							=generate_large_matrix()
_lowercase							:      Optional[int]      							=(
    [[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 A__  (			lowercase:     list[list[int]]					)				->		None:
       assert all(row == sorted(lowercase,		reverse=lowercase					) for row in grid					)
       assert all(list(lowercase					) == sorted(lowercase,		reverse=lowercase					) for col in zip(*lowercase					)					)



def A__  (			lowercase:     list[int]					)				->		int:
       A						:     Optional[Any]         =0
       A						:     List[Any]         =len(lowercase					) - 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:
              A						:     List[Any]         =(left + right) // 2
              A						:     Optional[int]         =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:
                     A						:     List[str]         =mid + 1
              else:
                     A						:     int         =mid - 1
    # No negative numbers so return the last index of the array + 1 which is the length.
       return len(lowercase					)



def A__  (			lowercase:     list[list[int]]					)				->		int:
       A						:     Optional[int]         =0
       A						:     Dict         =len(grid[0]					)

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



def A__  (			lowercase:     list[list[int]]					)				->		int:
       return len([number for row in grid for number in row if number < 0]					)



def A__  (			lowercase:     list[list[int]]					)				->		int:
       A						:     str         =0
       for row in grid:
              for i, number in enumerate(lowercase					):
                     if number < 0:
                            total += len(lowercase					) - i
                            break
       return total



def A__  (			)				->		None:
       from timeit import timeit

       print('Running benchmarks'					)
       A						:     Any         =(
           '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
       ):
              A						:     Optional[int]         =timeit(F'{func}(grid=grid)',		setup=lowercase,		number=500					)
              print(F'{func}() took {time:0.4f} seconds'					)


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    benchmark()







import heapq



def A__  (			lowercase:     dict					)				->		set[int]:
       A						:     list[list]         =[]

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

       # chosen_vertices = set of chosen vertices
       A						:     Dict         =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
              A						:     List[str]         =heapq.heappop(lowercase					)[1][0]
              chosen_vertices.add(lowercase					)

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


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    _lowercase							:      List[Any]      							={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)}''')






