




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



def       UpperCAmelCase       (     _snake_case						, _snake_case						, _snake_case						, _snake_case							):
							lowerCAmelCase   =		[[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_:
																												lowerCAmelCase   =		max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]						, dp[i - 1][w_]							)
																					else:
																												lowerCAmelCase   =		dp[i - 1][w_]

							return dp[n][w_], dp



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

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

							lowerCAmelCase   ,     lowerCAmelCase   =		knapsack(_snake_case						, _snake_case						, _snake_case						, _snake_case							)
							lowerCAmelCase   =		set()
							_construct_solution(_snake_case						, _snake_case						, _snake_case						, _snake_case						, _snake_case							)

							return optimal_val, example_optional_set



def       UpperCAmelCase       (     _snake_case						, _snake_case						, _snake_case						, _snake_case						, _snake_case							):
							# 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(_snake_case						, _snake_case						, i - 1						, _snake_case						, _snake_case							)
														else:
																					optimal_set.add(_snake_case							)
																					_construct_solution(_snake_case						, _snake_case						, i - 1						, j - wt[i - 1]						, _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_,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)









# DISCLAIMER: This file is strongly influenced by https://github.com/yang-song/score_sde_pytorch

import math
from typing import Union

import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import randn_tensor
from .scheduling_utils import SchedulerMixin


class 							__UpperCamelCase			( __UpperCAmelCase     ,					__UpperCAmelCase  ):





				'''simple docstring'''



				__a							: Any										=1



				@register_to_config
				def __init__( self       , UpperCAmelCase_=20_00       , UpperCAmelCase_=0.1       , UpperCAmelCase_=20       , UpperCAmelCase_=1E-3						):
											lowerCAmelCase   =		None
											lowerCAmelCase   =		None
											lowerCAmelCase   =		None



				def       __snake_case		( self       , UpperCAmelCase_       , UpperCAmelCase_ = None						):
											lowerCAmelCase   =		torch.linspace(1       , self.config.sampling_eps       , UpperCAmelCase_       , device=UpperCAmelCase_						)



				def       __snake_case		( self       , UpperCAmelCase_       , UpperCAmelCase_       , UpperCAmelCase_       , UpperCAmelCase_=None						):
											if self.timesteps is None:
																		raise ValueError(
																		    '''`self.timesteps` is not set, you need to run \'set_timesteps\' after creating the scheduler'''						)

											# TODO(Patrick) better comments + non-PyTorch
											# postprocess model score
											lowerCAmelCase   =		(
											    -0.25 * t**2 * (self.config.beta_max - self.config.beta_min) - 0.5 * t * self.config.beta_min
											)
											lowerCAmelCase   =		torch.sqrt(1.0 - torch.exp(2.0 * log_mean_coeff						)						)
											lowerCAmelCase   =		std.flatten()
											while len(std.shape						) < len(score.shape						):
																		lowerCAmelCase   =		std.unsqueeze(-1						)
											lowerCAmelCase   =		-score / std

											# compute
											lowerCAmelCase   =		-1.0 / len(self.timesteps						)

											lowerCAmelCase   =		self.config.beta_min + t * (self.config.beta_max - self.config.beta_min)
											lowerCAmelCase   =		beta_t.flatten()
											while len(beta_t.shape						) < len(x.shape						):
																		lowerCAmelCase   =		beta_t.unsqueeze(-1						)
											lowerCAmelCase   =		-0.5 * beta_t * x

											lowerCAmelCase   =		torch.sqrt(UpperCAmelCase_						)
											lowerCAmelCase   =		drift - diffusion**2 * score
											lowerCAmelCase   =		x + drift * dt

											# add noise
											lowerCAmelCase   =		randn_tensor(x.shape       , layout=x.layout       , generator=UpperCAmelCase_       , device=x.device       , dtype=x.dtype						)
											lowerCAmelCase   =		x_mean + diffusion * math.sqrt(-dt						) * noise

											return x, x_mean






				def __len__( self						):
											return self.config.num_train_timesteps



