





'''simple docstring'''







def        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			List[str],			SCREAMING_SNAKE_CASE__      :			Optional[Any],			SCREAMING_SNAKE_CASE__      :			Optional[Any],			SCREAMING_SNAKE_CASE__      :			str   ) ->	Any:
						global f  # a global dp table for knapsack
						if f[i][j] < 0:
												if j < wt[i - 1]:
																		UpperCAmelCase_			:      List[Any]     						=  mf_knapsack(i - 1,			SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   )
												else:
																		UpperCAmelCase_			:      Tuple     						=  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],			)
												UpperCAmelCase_			:      Tuple     						=  val
						return f[i][j]





def        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			Dict,			SCREAMING_SNAKE_CASE__      :			int,			SCREAMING_SNAKE_CASE__      :			Union[str, Any],			SCREAMING_SNAKE_CASE__      :			Any   ) ->	Dict:
						UpperCAmelCase_			:      Optional[Any]     						=  [[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_			:      List[str]     						=  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        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			int,			SCREAMING_SNAKE_CASE__      :			list,			SCREAMING_SNAKE_CASE__      :			list   ) ->	Optional[int]:
						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'''   )

						UpperCAmelCase_			:      List[Any]     						=  len(SCREAMING_SNAKE_CASE__   )
						if num_items != len(SCREAMING_SNAKE_CASE__   ):
												UpperCAmelCase_			:      Union[str, Any]     						=  (
												    '''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__   ):
																		UpperCAmelCase_			:      Union[str, Any]     						=  (
																		    '''All weights must be integers but got weight of '''
																		    F"""type {type(wt[i]   )} at index {i}"""
																		)
																		raise TypeError(SCREAMING_SNAKE_CASE__   )

						UpperCAmelCase_						,       UpperCAmelCase_			:      str     						=  knapsack(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   )
						UpperCAmelCase_			:      set     						=  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        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			list,			SCREAMING_SNAKE_CASE__      :			list,			SCREAMING_SNAKE_CASE__      :			int,			SCREAMING_SNAKE_CASE__      :			int,			SCREAMING_SNAKE_CASE__      :			set   ) ->	Tuple:
						# 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(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__":
					snake_case_     :       List[str]						=  [3, 2, 4, 4]
					snake_case_     :       Optional[int]						=  [4, 3, 2, 3]
					snake_case_     :       List[Any]						=  4
					snake_case_     :       Union[str, Any]						=  6
					snake_case_     :       Dict						=  [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
					snake_case_			,snake_case_     :       Tuple						=  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
					snake_case_			,snake_case_     :       Any						=  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'''







import sys
import turtle





def        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			tuple[float, float],			SCREAMING_SNAKE_CASE__      :			tuple[float, float]   ) ->	tuple[float, float]:
						return (pa[0] + pa[0]) / 2, (pa[1] + pa[1]) / 2





def        lowerCamelCase_      (  SCREAMING_SNAKE_CASE__      :			tuple[float, float],			SCREAMING_SNAKE_CASE__      :			tuple[float, float],			SCREAMING_SNAKE_CASE__      :			tuple[float, float],			SCREAMING_SNAKE_CASE__      :			int,			) ->	None:
						my_pen.up()
						my_pen.goto(vertexa[0],			vertexa[1]   )
						my_pen.down()
						my_pen.goto(vertexa[0],			vertexa[1]   )
						my_pen.goto(vertexa[0],			vertexa[1]   )
						my_pen.goto(vertexa[0],			vertexa[1]   )

						if depth == 0:
												return

						triangle(SCREAMING_SNAKE_CASE__,			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			depth - 1   )
						triangle(SCREAMING_SNAKE_CASE__,			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			depth - 1   )
						triangle(SCREAMING_SNAKE_CASE__,			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			get_mid(SCREAMING_SNAKE_CASE__,			SCREAMING_SNAKE_CASE__   ),			depth - 1   )


if __name__ == "__main__":
					if len(sys.argv) != 2:
										raise ValueError(
										    "Correct format for using this script: "
										    "python fractals.py <int:depth_for_fractal>"
										)
					snake_case_     :       Any						=  turtle.Turtle()
					my_pen.ht()
					my_pen.speed(5)
					my_pen.pencolor("red")

					snake_case_     :       Tuple						=  [(-1_75, -1_25), (0, 1_75), (1_75, -1_25)]  # vertices of triangle
					triangle(vertices[0], vertices[1], vertices[2], int(sys.argv[1]))





