

import heapq



def       __lowerCAmelCase      (						A    ):
	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(lowerCAmelCase__       ,     [-1 * len(lowerCAmelCase__    ), (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(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]:
				UpperCAmelCase_       				=			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()

				_a:    List[str] 							=			{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)}')







def       __lowerCAmelCase      (						A    ):
	UpperCAmelCase_       				=			generate_pascal_triangle(A    )
	for row_idx in range(A    ):
		# Print left spaces
		for _ in range(num_rows - row_idx - 1    ):
			print(end=" "    )
		# Print row values
		for col_idx in range(row_idx + 1    ):
			if col_idx != row_idx:
				print(triangle[row_idx][col_idx]       ,     end=" "    )
			else:
				print(triangle[row_idx][col_idx]       ,     end=""    )
		print()



def       __lowerCAmelCase      (						A    ):

	if not isinstance(A       ,     A    ):
		raise TypeError("The input value of 'num_rows' should be 'int'"    )

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    "The input value of 'num_rows' should be greater than or equal to 0"    )

	UpperCAmelCase_       				=			[]
	for current_row_idx in range(A    ):
		UpperCAmelCase_       				=			populate_current_row(A       ,     A    )
		triangle.append(A    )
	return triangle



def       __lowerCAmelCase      (						A       ,     A    ):
	UpperCAmelCase_       				=			[-1] * (current_row_idx + 1)
	# first and last elements of current row are equal to 1
	UpperCAmelCase_				,   UpperCAmelCase_       				=			1, 1
	for current_col_idx in range(1       ,     A    ):
		calculate_current_element(
		    A       ,     A       ,     A       ,     A    )
	return current_row



def       __lowerCAmelCase      (						A       ,     A       ,     A       ,     A       ,     ):
	UpperCAmelCase_       				=			triangle[current_row_idx - 1][current_col_idx - 1]
	UpperCAmelCase_       				=			triangle[current_row_idx - 1][current_col_idx]
	UpperCAmelCase_       				=			above_to_left_elt + above_to_right_elt



def       __lowerCAmelCase      (						A    ):

	if not isinstance(A       ,     A    ):
		raise TypeError("The input value of 'num_rows' should be 'int'"    )

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    "The input value of 'num_rows' should be greater than or equal to 0"    )

	UpperCAmelCase_       				=			[[1]]

	for row_index in range(1       ,     A    ):
		UpperCAmelCase_       				=			[0] + result[-1] + [0]
		UpperCAmelCase_       				=			row_index + 1
		# Calculate the number of distinct elements in a row
		UpperCAmelCase_       				=			sum(divmod(A       ,     2    )    )
		UpperCAmelCase_       				=			[
		    temp_row[i - 1] + temp_row[i] for i in range(1       ,     distinct_elements + 1    )
		]
		UpperCAmelCase_       				=			row_first_half[: (row_index + 1) // 2]
		row_second_half.reverse()
		UpperCAmelCase_       				=			row_first_half + row_second_half
		result.append(A    )

	return result



def       __lowerCAmelCase      (						):
	from collections.abc import Callable
	from timeit import timeit

	def benchmark_a_function(A       ,     A    ) -> None:
		UpperCAmelCase_       				=			F"{func.__name__}({value})"
		UpperCAmelCase_       				=			timeit(F"__main__.{call}"       ,     setup="import __main__"    )
		# print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
		print(F"{call:38} -- {timing:.4f} seconds"    )

	for value in range(15    ):  # (1, 7, 14):
		for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
			benchmark_a_function(A       ,     A    )
		print()


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				benchmark()