



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






def     __a					(		__UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase       ):
 a__												= [[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_:
    a__												= max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]							,  dp[i - 1][w_]       )
   else:
    a__												= dp[i - 1][w_]

 return dp[n][w_], dp






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

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

 a__							,			a__												= knapsack(__UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase       )
 a__												= set()
 _construct_solution(__UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase       )

 return optimal_val, example_optional_set






def     __a					(		__UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase							,  __UpperCAmelCase       ):
 # 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(__UpperCAmelCase							,  __UpperCAmelCase							,  i - 1							,  __UpperCAmelCase							,  __UpperCAmelCase       )
  else:
   optimal_set.add(__UpperCAmelCase       )
   _construct_solution(__UpperCAmelCase							,  __UpperCAmelCase							,  i - 1							,  j - wt[i - 1]							,  __UpperCAmelCase       )


if __name__ == "__main__":
     a_      :      Dict    =     [3, 2, 4, 4]
     a_      :      List[str]    =     [4, 3, 2, 3]
     a_      :      Any    =     4
     a_      :      Optional[Any]    =     6
     a_      :      Union[str, Any]    =     [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
     a_ ,     a_      :      List[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
     a_ ,     a_      :      Optional[int]    =     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)










import math
import time
from typing import Dict, List, Optional

from torch.utils.data import Dataset

from transformers import SeqaSeqTrainer, is_torch_tpu_available
from transformers.trainer_utils import PredictionOutput, speed_metrics


if is_torch_tpu_available(check_device=False):
     import torch_xla.core.xla_model as xm
     import torch_xla.debug.metrics as met
class    __UpperCamelCase			(       _lowercase				):

 """simple docstring"""



 def __init__(	self		,				*SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE=None		,				SCREAMING_SNAKE_CASE=None		,				**SCREAMING_SNAKE_CASE      ) ->			Tuple:
  super().__init__(*SCREAMING_SNAKE_CASE		,				**SCREAMING_SNAKE_CASE      )
  a__												= eval_examples
  a__												= post_process_function

 def 						_UpperCAmelCase       (	self		,				SCREAMING_SNAKE_CASE = None		,				SCREAMING_SNAKE_CASE=None		,				SCREAMING_SNAKE_CASE = None		,				SCREAMING_SNAKE_CASE = "eval"		,				**SCREAMING_SNAKE_CASE		,				) ->			Dict[str, float]:
  a__												= gen_kwargs.copy()
  a__												= (
      gen_kwargs['''max_length'''] if gen_kwargs.get('''max_length'''      ) is not None else self.args.generation_max_length
  )
  a__												= (
      gen_kwargs['''num_beams'''] if gen_kwargs.get('''num_beams'''      ) is not None else self.args.generation_num_beams
  )
  a__												= gen_kwargs

  a__												= self.eval_dataset if eval_dataset is None else eval_dataset
  a__												= self.get_eval_dataloader(SCREAMING_SNAKE_CASE      )
  a__												= self.eval_examples if eval_examples is None else eval_examples

  # Temporarily disable metric computation, we will do it in the loop here.
  a__												= self.compute_metrics
  a__												= None
  a__												= time.time()
  a__												= self.prediction_loop if self.args.use_legacy_prediction_loop else self.evaluation_loop
  try:
   a__												= eval_loop(
       SCREAMING_SNAKE_CASE		,				description='''Evaluation'''		,				prediction_loss_only=True if compute_metrics is None else None		,				ignore_keys=SCREAMING_SNAKE_CASE		,				metric_key_prefix=SCREAMING_SNAKE_CASE		,				)
  finally:
   a__												= compute_metrics
  a__												= self.args.eval_batch_size * self.args.world_size
  if f"{metric_key_prefix}_jit_compilation_time" in output.metrics:
   start_time += output.metrics[f"{metric_key_prefix}_jit_compilation_time"]
  output.metrics.update(
      speed_metrics(
          SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				num_samples=output.num_samples		,				num_steps=math.ceil(output.num_samples / total_batch_size      )		,				)      )

  if self.post_process_function is not None and self.compute_metrics is not None and self.args.should_save:
   # Only the main node write the results by default
   a__												= self.post_process_function(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )
   a__												= self.compute_metrics(SCREAMING_SNAKE_CASE      )

   # Prefix all keys with metric_key_prefix + '_'
   for key in list(metrics.keys()      ):
    if not key.startswith(f"{metric_key_prefix}_"      ):
     a__												= metrics.pop(SCREAMING_SNAKE_CASE      )

   metrics.update(output.metrics      )
  else:
   a__												= output.metrics

  if self.args.should_log:
   # Only the main node log the results by default
   self.log(SCREAMING_SNAKE_CASE      )

  if self.args.tpu_metrics_debug or self.args.debug:
   # tpu-comment: Logging debug metrics for PyTorch/XLA (compile, execute times, ops, etc.)
   xm.master_print(met.metrics_report()      )

  a__												= self.callback_handler.on_evaluate(self.args		,				self.state		,				self.control		,				SCREAMING_SNAKE_CASE      )
  return metrics



 def 						_UpperCAmelCase       (	self		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE=None		,				SCREAMING_SNAKE_CASE = "test"		,				**SCREAMING_SNAKE_CASE      ) ->			List[str]:
  a__												= gen_kwargs.copy()

  a__												= self.get_test_dataloader(SCREAMING_SNAKE_CASE      )

  # Temporarily disable metric computation, we will do it in the loop here.
  a__												= self.compute_metrics
  a__												= None
  a__												= time.time()
  a__												= self.prediction_loop if self.args.use_legacy_prediction_loop else self.evaluation_loop
  try:
   a__												= eval_loop(
       SCREAMING_SNAKE_CASE		,				description='''Prediction'''		,				prediction_loss_only=True if compute_metrics is None else None		,				ignore_keys=SCREAMING_SNAKE_CASE		,				metric_key_prefix=SCREAMING_SNAKE_CASE		,				)
  finally:
   a__												= compute_metrics

  a__												= self.args.eval_batch_size * self.args.world_size
  if f"{metric_key_prefix}_jit_compilation_time" in output.metrics:
   start_time += output.metrics[f"{metric_key_prefix}_jit_compilation_time"]
  output.metrics.update(
      speed_metrics(
          SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				num_samples=output.num_samples		,				num_steps=math.ceil(output.num_samples / total_batch_size      )		,				)      )
  if self.post_process_function is None or self.compute_metrics is None:
   return output

  a__												= self.post_process_function(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				'''predict'''      )
  a__												= self.compute_metrics(SCREAMING_SNAKE_CASE      )

  # Prefix all keys with metric_key_prefix + '_'
  for key in list(metrics.keys()      ):
   if not key.startswith(f"{metric_key_prefix}_"      ):
    a__												= metrics.pop(SCREAMING_SNAKE_CASE      )
  metrics.update(output.metrics      )
  return PredictionOutput(predictions=predictions.predictions		,				label_ids=predictions.label_ids		,				metrics=SCREAMING_SNAKE_CASE      )





