


def 			lowerCAmelCase (		lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ )-> Dict:
				global f  # a global dp table for knapsack
				if f[i][j] < 0:
								if j < wt[i - 1]:
												lowerCAmelCase_ :      Tuple    				=				mf_knapsack(i - 1 ,   a__ ,   a__ ,   a__ )
								else:
												lowerCAmelCase_ :      Optional[int]    				=				max(
												    mf_knapsack(i - 1 ,   a__ ,   a__ ,   a__ ) ,   mf_knapsack(i - 1 ,   a__ ,   a__ ,   j - wt[i - 1] ) + val[i - 1] ,   )
								lowerCAmelCase_ :      int    				=				val
				return f[i][j]



def 			lowerCAmelCase (		lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ )-> Union[str, Any]:
				lowerCAmelCase_ :      List[str]    				=				[[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_ :      Dict    				=				max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]] ,   dp[i - 1][w_] )
												else:
																lowerCAmelCase_ :      str    				=				dp[i - 1][w_]

				return dp[n][w_], dp



def 			lowerCAmelCase (		lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ )-> Dict:

				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''' )

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

				lowerCAmelCase_				,  lowerCAmelCase_ :      List[str]    				=				knapsack(a__ ,   a__ ,   a__ ,   a__ )
				lowerCAmelCase_ :      Optional[int]    				=				set()
				_construct_solution(a__ ,   a__ ,   a__ ,   a__ ,   a__ )

				return optimal_val, example_optional_set



def 			lowerCAmelCase (		lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ ,   lowerCAmelCase_ )-> Dict:

				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__":
						_UpperCAmelCase			:  Optional[Any]     =[3, 2, 4, 4]
						_UpperCAmelCase			:  Tuple     =[4, 3, 2, 3]
						_UpperCAmelCase			:  Optional[Any]     =4
						_UpperCAmelCase			:  List[Any]     =6
						_UpperCAmelCase			:  int     =[[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
						_UpperCAmelCase     ,					_UpperCAmelCase			:  Union[str, Any]     =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			:  Union[str, 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)








# We ignore warnings about stepping the scheduler since we step it ourselves during gradient accumulation

import warnings

from .state import AcceleratorState, GradientState


warnings.filterwarnings("""ignore""", category=UserWarning, module="""torch.optim.lr_scheduler""")







class 				snake_case__:







						'''simple docstring'''




						def __init__(				self   ,						__lowercase   ,						__lowercase   ,						__lowercase = True   ,						__lowercase = False							)					->    Tuple:
										lowerCAmelCase_ :      Optional[int]    				=				scheduler
										lowerCAmelCase_ :      Dict    				=				optimizers if isinstance(__lowercase   ,						(list, tuple)							) else [optimizers]
										lowerCAmelCase_ :      str    				=				split_batches
										lowerCAmelCase_ :      Any    				=				step_with_optimizer
										lowerCAmelCase_ :      Optional[Any]    				=				GradientState()




						def 						lowercase_ (				self   ,						*__lowercase   ,						**__lowercase							)					->    Any:
										if not self.step_with_optimizer:
														# No link between scheduler and optimizer -> just step
														self.scheduler.step(*__lowercase   ,						**__lowercase							)
														return

										# Otherwise, first make sure the optimizer was stepped.
										if not self.gradient_state.sync_gradients:
														if self.gradient_state.adjust_scheduler:
																		self.scheduler._step_count += 1
														return

										for opt in self.optimizers:
														if opt.step_was_skipped:
																		return
										if self.split_batches:
														# Split batches -> the training dataloader batch size is not changed so one step per training step
														self.scheduler.step(*__lowercase   ,						**__lowercase							)
										else:
														# Otherwise the training dataloader batch size was multiplied by `num_processes`, so we need to do
														# num_processes steps per training step
														lowerCAmelCase_ :      Optional[Any]    				=				AcceleratorState().num_processes
														for _ in range(__lowercase							):
																		# Special case when using OneCycle and `drop_last` was not used
																		if hasattr(self.scheduler   ,						'''total_steps'''							):
																						if self.scheduler._step_count <= self.scheduler.total_steps:
																										self.scheduler.step(*__lowercase   ,						**__lowercase							)
																		else:
																						self.scheduler.step(*__lowercase   ,						**__lowercase							)




						def 						lowercase_ (				self							)					->    Union[str, Any]:
										return self.scheduler.get_last_lr()




						def 						lowercase_ (				self							)					->    List[str]:
										return self.scheduler.state_dict()




						def 						lowercase_ (				self   ,						__lowercase							)					->    int:
										self.scheduler.load_state_dict(__lowercase							)




						def 						lowercase_ (				self							)					->    Tuple:
										return self.scheduler.get_lr()




						def 						lowercase_ (				self   ,						*__lowercase   ,						**__lowercase							)					->    int:
										return self.scheduler.print_lr(*__lowercase   ,						**__lowercase							)