
def    a      ( a		,						a		,						a		,						a	)     ->List[str]:




			'''simple docstring'''


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


def    a      ( a		,						a		,						a		,						a	)     ->Tuple:




			'''simple docstring'''


			SCREAMING_SNAKE_CASE     						= [[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_:
												SCREAMING_SNAKE_CASE     						= max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]		,						dp[i - 1][w_]	)
									else:
												SCREAMING_SNAKE_CASE     						= dp[i - 1][w_]

			return dp[n][w_], dp


def    a      ( a		,						a		,						a	)     ->List[Any]:




			'''simple docstring'''


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

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

			SCREAMING_SNAKE_CASE						,							SCREAMING_SNAKE_CASE     						= knapsack(a		,						a		,						a		,						a	)
			SCREAMING_SNAKE_CASE     						= set()
			_construct_solution(a		,						a		,						a		,						a		,						a	)

			return optimal_val, example_optional_set


def    a      ( a		,						a		,						a		,						a		,						a	)     ->Union[str, Any]:




			'''simple docstring'''


			if i > 0 and j > 0:
						if dp[i - 1][j] == dp[i][j]:
									_construct_solution(a		,						a		,						i - 1		,						a		,						a	)
						else:
									optimal_set.add(a	)
									_construct_solution(a		,						a		,						i - 1		,						j - wt[i - 1]		,						a	)


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





from __future__ import annotations

from decimal import Decimal

from numpy import array

def lowercase    (  SCREAMING_SNAKE_CASE )			->							list[list[float]]:


			'''simple docstring'''







			SCREAMING_SNAKE_CASE_								=						Decimal

			# Check if the provided matrix has 2 rows and 2 columns
			# since this implementation only works for 2x2 matrices
			if len(SCREAMING_SNAKE_CASE ) == 2 and len(matrix[0] ) == 2 and len(matrix[1] ) == 2:
						# Calculate the determinant of the matrix
						SCREAMING_SNAKE_CASE_								=						float(
						    d(matrix[0][0] ) * d(matrix[1][1] ) - d(matrix[1][0] ) * d(matrix[0][1] ) )
						if determinant == 0:
									raise ValueError('This matrix has no inverse.' )

						# Creates a copy of the matrix with swapped positions of the elements
						SCREAMING_SNAKE_CASE_								=						[[0.0, 0.0], [0.0, 0.0]]
						SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_								=						matrix[1][1], matrix[0][0]
						SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_								=						-matrix[1][0], -matrix[0][1]

						# Calculate the inverse of the matrix
						return [
						    [(float(d(SCREAMING_SNAKE_CASE ) ) / determinant) or 0.0 for n in row] for row in swapped_matrix
						]
			elif (
			    len(SCREAMING_SNAKE_CASE ) == 3
			    and len(matrix[0] ) == 3
			    and len(matrix[1] ) == 3
			    and len(matrix[2] ) == 3
			):
						# Calculate the determinant of the matrix using Sarrus rule
						SCREAMING_SNAKE_CASE_								=						float(
						    (
						        (d(matrix[0][0] ) * d(matrix[1][1] ) * d(matrix[2][2] ))
						        + (d(matrix[0][1] ) * d(matrix[1][2] ) * d(matrix[2][0] ))
						        + (d(matrix[0][2] ) * d(matrix[1][0] ) * d(matrix[2][1] ))
						    )
						    - (
						        (d(matrix[0][2] ) * d(matrix[1][1] ) * d(matrix[2][0] ))
						        + (d(matrix[0][1] ) * d(matrix[1][0] ) * d(matrix[2][2] ))
						        + (d(matrix[0][0] ) * d(matrix[1][2] ) * d(matrix[2][1] ))
						    ) )
						if determinant == 0:
									raise ValueError('This matrix has no inverse.' )

						# Creating cofactor matrix
						SCREAMING_SNAKE_CASE_								=						[
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						]
						SCREAMING_SNAKE_CASE_								=						(d(matrix[1][1] ) * d(matrix[2][2] )) - (
						    d(matrix[1][2] ) * d(matrix[2][1] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[1][0] ) * d(matrix[2][2] )) - (d(matrix[1][2] ) * d(matrix[2][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[1][0] ) * d(matrix[2][1] )) - (
						    d(matrix[1][1] ) * d(matrix[2][0] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][1] ) * d(matrix[2][2] )) - (d(matrix[0][2] ) * d(matrix[2][1] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][0] ) * d(matrix[2][2] )) - (
						    d(matrix[0][2] ) * d(matrix[2][0] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][0] ) * d(matrix[2][1] )) - (d(matrix[0][1] ) * d(matrix[2][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][1] ) * d(matrix[1][2] )) - (
						    d(matrix[0][2] ) * d(matrix[1][1] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][0] ) * d(matrix[1][2] )) - (d(matrix[0][2] ) * d(matrix[1][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][0] ) * d(matrix[1][1] )) - (
						    d(matrix[0][1] ) * d(matrix[1][0] )
						)

						# Transpose the cofactor matrix (Adjoint matrix)
						SCREAMING_SNAKE_CASE_								=						array(SCREAMING_SNAKE_CASE )
						for i in range(3 ):
									for j in range(3 ):
												SCREAMING_SNAKE_CASE_								=						cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
						SCREAMING_SNAKE_CASE_								=						array(SCREAMING_SNAKE_CASE )
						for i in range(3 ):
									for j in range(3 ):
												inverse_matrix[i][j] /= d(SCREAMING_SNAKE_CASE )

        # Calculate the inverse of the matrix
						return [[float(d(SCREAMING_SNAKE_CASE ) ) or 0.0 for n in row] for row in inverse_matrix]
			raise ValueError('Please provide a matrix of size 2x2 or 3x3.' )
