



def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__						,       lowercase__   ) ->      Dict:
						global f  # a global dp table for knapsack
						if f[i][j] < 0:
												if j < wt[i - 1]:
																		__lowercase					=				mf_knapsack(i - 1						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE   )
												else:
																		__lowercase					=				max(
																		    mf_knapsack(i - 1						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE   )						,       mf_knapsack(i - 1						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       j - wt[i - 1]   ) + val[i - 1]						,       )
												__lowercase					=				val
						return f[i][j]






def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__						,       lowercase__   ) ->      Optional[int]:
						__lowercase					=				[[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_:
																								__lowercase					=				max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]						,       dp[i - 1][w_]   )
																		else:
																								__lowercase					=				dp[i - 1][w_]

						return dp[n][w_], dp






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



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

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

						__lowercase	,			__lowercase					=				knapsack(_SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE   )
						__lowercase					=				set()
						_construct_solution(_SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE						,       _SCREAMING_SNAKE_CASE   )

						return optimal_val, example_optional_set






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



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


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__    =						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__    =						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 collections.abc import Callable

UpperCamelCase__ =						list[list[float | int]]






def  UpperCAmelCase__    (    lowercase__						,       lowercase__   ) ->      Matrix:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(size + 1   )] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for row in range(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				matrix[row][col]

												__lowercase					=				vector[row][0]

						__lowercase					=				0
						__lowercase					=				0
						while row < size and col < size:
												# pivoting
												__lowercase					=				max((abs(augmented[rowa][col]   ), rowa) for rowa in range(lowercase__						,       lowercase__   )   )[
												    1
												]
												if augmented[pivot_row][col] == 0:
																		col += 1
																		continue
												else:
																		__lowercase	,			__lowercase					=				augmented[pivot_row], augmented[row]

												for rowa in range(row + 1						,       lowercase__   ):
																		__lowercase					=				augmented[rowa][col] / augmented[row][col]
																		__lowercase					=				0
																		for cola in range(col + 1						,       size + 1   ):
																								augmented[rowa][cola] -= augmented[row][cola] * ratio

												row += 1
												col += 1

						# back substitution
						for col in range(1						,       lowercase__   ):
												for row in range(lowercase__   ):
																		__lowercase					=				augmented[row][col] / augmented[col][col]
																		for cola in range(lowercase__						,       size + 1   ):
																								augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
						return [
						    [round(augmented[row][size] / augmented[row][row]						,       10   )] for row in range(lowercase__   )
						]






def  UpperCAmelCase__    (    lowercase__   ) ->      Callable[[int], int]:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(lowercase__   )] for _ in range(lowercase__   )]
						__lowercase					=				[[0] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for x_val, y_val in enumerate(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				(x_val + 1) ** (size - col - 1)
												__lowercase					=				y_val

						__lowercase					=				solve(lowercase__						,       lowercase__   )

						def interpolated_func(lowercase__   ) -> int:
												return sum(
												    round(coeffs[x_val][0]   ) * (var ** (size - x_val - 1))
												    for x_val in range(lowercase__   )   )

						return interpolated_func






def  UpperCAmelCase__    (    lowercase__   ) ->      int:
						return (
						    1
						    - variable
						    + variable**2
						    - variable**3
						    + variable**4
						    - variable**5
						    + variable**6
						    - variable**7
						    + variable**8
						    - variable**9
						    + variable**10
						)






def  UpperCAmelCase__    (    lowercase__ = question_function						,       lowercase__ = 10   ) ->      int:
						__lowercase					=				[func(lowercase__   ) for x_val in range(1						,       order + 1   )]

						__lowercase					=				[
						    interpolate(data_points[:max_coeff]   ) for max_coeff in range(1						,       order + 1   )
						]

						__lowercase					=				0
						__lowercase					=				42
						__lowercase					=				42

						for poly in polynomials:
												__lowercase					=				1
												while func(lowercase__   ) == poly(lowercase__   ):
																		x_val += 1

												ret += poly(lowercase__   )

						return ret


if __name__ == "__main__":
			print(F"""{solution() = }""")





