



'''simple docstring'''






def 				snake_case_  (      _lowerCAmelCase						:		Optional[int]     ,   _lowerCAmelCase						:		int     ,   _lowerCAmelCase						:		Any     ,   _lowerCAmelCase						:		Dict						)   ->     List[Any]:
							global f  # a global dp table for knapsack
							if f[i][j] < 0:
														if j < wt[i - 1]:
																					UpperCAmelCase				: Tuple											=	mf_knapsack(i - 1     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase						)
														else:
																					UpperCAmelCase				: List[Any]											=	max(
																					    mf_knapsack(i - 1     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase						)     ,   mf_knapsack(i - 1     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   j - wt[i - 1]						) + val[i - 1]     ,   )
														UpperCAmelCase				: List[str]											=	val
							return f[i][j]


def 				snake_case_  (      _lowerCAmelCase						:		List[Any]     ,   _lowerCAmelCase						:		Optional[int]     ,   _lowerCAmelCase						:		Tuple     ,   _lowerCAmelCase						:		Optional[int]						)   ->     int:
							UpperCAmelCase				: List[str]											=	[[0] * (w + 1) for _ in range(n + 1						)]

							for i in range(1     ,   n + 1						):
														for w_ in range(1     ,   w + 1						):
																					if wt[i - 1] <= w_:
																												UpperCAmelCase				: Optional[int]											=	max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]     ,   dp[i - 1][w_]						)
																					else:
																												UpperCAmelCase				: List[str]											=	dp[i - 1][w_]

							return dp[n][w_], dp


def 				snake_case_  (      _lowerCAmelCase						:		int     ,   _lowerCAmelCase						:		list     ,   _lowerCAmelCase						:		list						)   ->     Tuple:
							if not (isinstance(_lowerCAmelCase     ,   (list, tuple)						) and isinstance(_lowerCAmelCase     ,   (list, tuple)						)):
														raise ValueError(
														    '''Both the weights and values vectors must be either lists or tuples'''						)

							UpperCAmelCase				: List[str]											=	len(_lowerCAmelCase						)
							if num_items != len(_lowerCAmelCase						):
														UpperCAmelCase				: List[str]											=	(
														    '''The number of weights must be the same as the number of values.\n'''
														    f"""But got {num_items} weights and {len(_lowerCAmelCase						)} values"""
														)
														raise ValueError(_lowerCAmelCase						)
							for i in range(_lowerCAmelCase						):
														if not isinstance(wt[i]     ,   _lowerCAmelCase						):
																					UpperCAmelCase				: Any											=	(
																					    '''All weights must be integers but got weight of '''
																					    f"""type {type(wt[i]						)} at index {i}"""
																					)
																					raise TypeError(_lowerCAmelCase						)

							UpperCAmelCase							,	UpperCAmelCase				: Optional[Any]											=	knapsack(_lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase						)
							UpperCAmelCase				: set											=	set()
							_construct_solution(_lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase						)

							return optimal_val, example_optional_set


def 				snake_case_  (      _lowerCAmelCase						:		list     ,   _lowerCAmelCase						:		list     ,   _lowerCAmelCase						:		int     ,   _lowerCAmelCase						:		int     ,   _lowerCAmelCase						:		set						)   ->     List[Any]:
							# for the current item i at a maximum weight j to be part of an optimal subset,
							# the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
							# where i - 1 means considering only the previous items at the given maximum weight
							if i > 0 and j > 0:
														if dp[i - 1][j] == dp[i][j]:
																					_construct_solution(_lowerCAmelCase     ,   _lowerCAmelCase     ,   i - 1     ,   _lowerCAmelCase     ,   _lowerCAmelCase						)
														else:
																					optimal_set.add(_lowerCAmelCase						)
																					_construct_solution(_lowerCAmelCase     ,   _lowerCAmelCase     ,   i - 1     ,   j - wt[i - 1]     ,   _lowerCAmelCase						)


if __name__ == "__main__":
						UpperCamelCase__:     Tuple										=    [3, 2, 4, 4]
						UpperCamelCase__:     Any										=    [4, 3, 2, 3]
						UpperCamelCase__:     str										=    4
						UpperCamelCase__:     Tuple										=    6
						UpperCamelCase__:     int										=    [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
						UpperCamelCase__			,      UpperCamelCase__:     Union[str, Any]										=    knapsack(w, wt, val, n)
						print(optimal_solution)
						print(mf_knapsack(n, wt, val, w))  # switched the n and w

						# testing the dynamic programming problem with example
						# the optimal subset for the above example are items 3 and 4
						UpperCamelCase__			,      UpperCamelCase__:     List[str]										=    knapsack_with_example_solution(w, wt, val)
						assert optimal_solution == 8
						assert optimal_subset == {3, 4}
						print("optimal_value = ", optimal_solution)
						print("An optimal subset corresponding to the optimal value", optimal_subset)










'''simple docstring'''






def 				snake_case_  (      _lowerCAmelCase						:		Tuple     ,   _lowerCAmelCase						:		Tuple     ,   _lowerCAmelCase						:		int     ,   _lowerCAmelCase						:		Tuple						)   ->     Union[str, Any]:
							# Return True if there is node that has not iterated.
							UpperCAmelCase				: Optional[Any]											=	[False] * len(_lowerCAmelCase						)
							UpperCAmelCase				: Dict											=	[]
							queue.append(_lowerCAmelCase						)
							UpperCAmelCase				: str											=	True

							while queue:
														UpperCAmelCase				: Optional[int]											=	queue.pop(0						)
														for ind in range(len(graph[u]						)						):
																					if visited[ind] is False and graph[u][ind] > 0:
																												queue.append(_lowerCAmelCase						)
																												UpperCAmelCase				: Union[str, Any]											=	True
																												UpperCAmelCase				: Optional[int]											=	u

							return visited[t]


def 				snake_case_  (      _lowerCAmelCase						:		Union[str, Any]     ,   _lowerCAmelCase						:		List[str]     ,   _lowerCAmelCase						:		List[str]						)   ->     Any:
							# This array is filled by BFS and to store path
							UpperCAmelCase				: Dict											=	[-1] * (len(_lowerCAmelCase						))
							UpperCAmelCase				: Any											=	0
							while bfs(_lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase						):
														UpperCAmelCase				: List[str]											=	float('''Inf'''						)
														UpperCAmelCase				: str											=	sink

														while s != source:
																					# Find the minimum value in select path
																					UpperCAmelCase				: int											=	min(_lowerCAmelCase     ,   graph[parent[s]][s]						)
																					UpperCAmelCase				: Tuple											=	parent[s]

														max_flow += path_flow
														UpperCAmelCase				: Optional[Any]											=	sink

														while v != source:
																					UpperCAmelCase				: Dict											=	parent[v]
																					graph[u][v] -= path_flow
																					graph[v][u] += path_flow
																					UpperCAmelCase				: Dict											=	parent[v]
							return max_flow


UpperCamelCase__:     Tuple										=    [
    [0, 16, 13, 0, 0, 0],
    [0, 0, 10, 12, 0, 0],
    [0, 4, 0, 0, 14, 0],
    [0, 0, 9, 0, 0, 20],
    [0, 0, 0, 7, 0, 4],
    [0, 0, 0, 0, 0, 0],
]

UpperCamelCase__			,      UpperCamelCase__:     int										=    0, 5




print(ford_fulkerson(graph, source, sink))





