



def 							A			(  _UpperCAmelCase       :     List[str]							,     _UpperCAmelCase       :     Dict							,     _UpperCAmelCase       :     List[str]							,     _UpperCAmelCase       :     List[Any]					)       ->				str:







			'''simple docstring'''
			global f  # a global dp table for knapsack
			if f[i][j] < 0:
						if j < wt[i - 1]:
									_UpperCAmelCase	 =     mf_knapsack(i - 1							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase					)
						else:
									_UpperCAmelCase	 =     max(
									    mf_knapsack(i - 1							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase					)							,     mf_knapsack(i - 1							,     _UpperCAmelCase							,     _UpperCAmelCase							,     j - wt[i - 1]					) + val[i - 1]							,     )
						_UpperCAmelCase	 =     val
			return f[i][j]





def 							A			(  _UpperCAmelCase       :     str							,     _UpperCAmelCase       :     Union[str, Any]							,     _UpperCAmelCase       :     Union[str, Any]							,     _UpperCAmelCase       :     Union[str, Any]					)       ->				int:







			'''simple docstring'''
			_UpperCAmelCase	 =     [[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	 =     max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]							,     dp[i - 1][w_]					)
									else:
												_UpperCAmelCase	 =     dp[i - 1][w_]

			return dp[n][w_], dp





def 							A			(  _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     list							,     _UpperCAmelCase       :     list					)       ->				Any:







			'''simple docstring'''
			if not (isinstance(_UpperCAmelCase							,     (list, tuple)					) and isinstance(_UpperCAmelCase							,     (list, tuple)					)):
						raise ValueError(
						    'Both the weights and values vectors must be either lists or tuples'					)

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

			_UpperCAmelCase ,  _UpperCAmelCase	 =     knapsack(_UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase					)
			_UpperCAmelCase	 =     set()
			_construct_solution(_UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase					)

			return optimal_val, example_optional_set





def 							A			(  _UpperCAmelCase       :     list							,     _UpperCAmelCase       :     list							,     _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     set					)       ->				str:







			'''simple docstring'''
			# 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(_UpperCAmelCase							,     _UpperCAmelCase							,     i - 1							,     _UpperCAmelCase							,     _UpperCAmelCase					)
						else:
									optimal_set.add(_UpperCAmelCase					)
									_construct_solution(_UpperCAmelCase							,     _UpperCAmelCase							,     i - 1							,     j - wt[i - 1]							,     _UpperCAmelCase					)


if __name__ == "__main__":
							UpperCAmelCase__     = [3, 2, 4, 4]
							UpperCAmelCase__     = [4, 3, 2, 3]
							UpperCAmelCase__     = 4
							UpperCAmelCase__     = 6
							UpperCAmelCase__     = [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
							UpperCAmelCase__     ,						UpperCAmelCase__     = 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__     = 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)





def 							A			(  _UpperCAmelCase       :     list					)       ->				list:







			'''simple docstring'''
			if len(_UpperCAmelCase					) <= 1:
						return lst

			_UpperCAmelCase	 =     1

			while i < len(_UpperCAmelCase					):
						if lst[i - 1] <= lst[i]:
									i += 1
						else:
									_UpperCAmelCase ,  _UpperCAmelCase	 =     lst[i], lst[i - 1]
									i -= 1
									if i == 0:
												_UpperCAmelCase	 =     1

			return lst


if __name__ == "__main__":
							UpperCAmelCase__     = input("Enter numbers separated by a comma:\n").strip()
							UpperCAmelCase__     = [int(item) for item in user_input.split(",")]
							print(gnome_sort(unsorted))
