





from queue import PriorityQueue
from typing import Any

import numpy as np



def       A_    (					_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       )						->			float | int:
 for nxt, d in graph[v]:
  if nxt in visited_forward:
   continue
  UpperCamelCase     : List[str]	     =       cst_fwd.get(_lowerCAmelCase							,       np.inf					)
  UpperCamelCase     : int	     =       cst_fwd[v] + d
  if new_cost_f < old_cost_f:
   queue.put((new_cost_f, nxt)					)
   UpperCamelCase     : Tuple	     =       new_cost_f
   UpperCamelCase     : List[str]	     =       v
  if nxt in visited_backward:
   if cst_fwd[v] + d + cst_bwd[nxt] < shortest_distance:
    UpperCamelCase     : Optional[Any]	     =       cst_fwd[v] + d + cst_bwd[nxt]
 return shortest_distance



def       A_    (					_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase					)						->			int:
 UpperCamelCase     : int	     =       -1

 UpperCamelCase     : Optional[Any]	     =       set()
 UpperCamelCase     : str	     =       set()
 UpperCamelCase     : str	     =       {source: 0}
 UpperCamelCase     : str	     =       {destination: 0}
 UpperCamelCase     : List[str]	     =       {source: None}
 UpperCamelCase     : Union[str, Any]	     =       {destination: None}
 UpperCamelCase     : PriorityQueue[Any]	     =       PriorityQueue()
 UpperCamelCase     : PriorityQueue[Any]	     =       PriorityQueue()

 UpperCamelCase     : List[str]	     =       np.inf

 queue_forward.put((0, source)					)
 queue_backward.put((0, destination)					)

 if source == destination:
  return 0

 while not queue_forward.empty() and not queue_backward.empty():
  UpperCamelCase     : Tuple	     =       queue_forward.get()
  visited_forward.add(_lowerCAmelCase					)

  UpperCamelCase     : List[Any]	     =       queue_backward.get()
  visited_backward.add(_lowerCAmelCase					)

  UpperCamelCase     : Optional[int]	     =       pass_and_relaxation(
      _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       )

  UpperCamelCase     : int	     =       pass_and_relaxation(
      _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       )

  if cst_fwd[v_fwd] + cst_bwd[v_bwd] >= shortest_distance:
   break

 if shortest_distance != np.inf:
  UpperCamelCase     : Optional[Any]	     =       shortest_distance
 return shortest_path_distance


__lowerCamelCase	:					Tuple			  =					{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""G""", 2]],
    """F""": [],
    """G""": [["""F""", 1]],
}
__lowerCamelCase	:					int			  =					{
    """B""": [["""E""", 1]],
    """C""": [["""B""", 1]],
    """D""": [["""C""", 1]],
    """F""": [["""D""", 1], ["""G""", 1]],
    """E""": [[None, np.inf]],
    """G""": [["""E""", 2]],
}

if __name__ == "__main__":
    import doctest

    doctest.testmod()







from typing import Callable, Dict, Optional, Tuple

import torch
from torch import nn
from torch.distributions import (
    AffineTransform,
    Distribution,
    Independent,
    NegativeBinomial,
    Normal,
    StudentT,
    TransformedDistribution,
)



class  A__       (	__snake_case      ):





      def __init__(			self	,							A_	,							A_=None	,							A_=None	,							A_=0       ):
       '''simple docstring'''




       UpperCamelCase     : Union[str, Any]	     =       1.0 if scale is None else scale
       UpperCamelCase     : Optional[int]	     =       0.0 if loc is None else loc

       super().__init__(A_	,							[AffineTransform(loc=self.loc	,							scale=self.scale	,							event_dim=A_       )]       )





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return self.base_dist.mean * self.scale + self.loc





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return self.base_dist.variance * self.scale**2





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return self.variance.sqrt()



class  A__       (	nn.Module      ):





      def __init__(			self	,							A_	,							A_	,							A_	,							**A_       ):
       '''simple docstring'''




       super().__init__(**A_       )
       UpperCamelCase     : Union[str, Any]	     =       args_dim
       UpperCamelCase     : str	     =       nn.ModuleList([nn.Linear(A_	,							A_       ) for dim in args_dim.values()]       )
       UpperCamelCase     : Union[str, Any]	     =       domain_map





      def 	__UpperCamelCase(			self	,							A_       ):
       '''simple docstring'''




       UpperCamelCase     : List[Any]	     =       [proj(A_       ) for proj in self.proj]

       return self.domain_map(*A_       )



class  A__       (	nn.Module      ):





      def __init__(			self	,							A_       ):
       '''simple docstring'''




       super().__init__()
       UpperCamelCase     : str	     =       function





      def 	__UpperCamelCase(			self	,							A_	,							*A_       ):
       '''simple docstring'''




       return self.function(A_	,							*A_       )



class  A__       :
      _UpperCAmelCase     :type
      _UpperCAmelCase     :int
      _UpperCAmelCase     :Dict[str, int]





      def __init__(			self	,							A_ = 1       ):
       '''simple docstring'''




       UpperCamelCase     : Tuple	     =       dim
       UpperCamelCase     : Union[str, Any]	     =       {k: dim * self.args_dim[k] for k in self.args_dim}





      def 	__UpperCamelCase(			self	,							A_       ):
       '''simple docstring'''




       if self.dim == 1:
        return self.distribution_class(*A_       )
       else:
        return Independent(self.distribution_class(*A_       )	,							1       )





      def 	__UpperCamelCase(			self	,							A_	,							A_ = None	,							A_ = None	,							):
       '''simple docstring'''




       UpperCamelCase     : str	     =       self._base_distribution(A_       )
       if loc is None and scale is None:
        return distr
       else:
        return AffineTransformed(A_	,							loc=A_	,							scale=A_	,							event_dim=self.event_dim       )





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return () if self.dim == 1 else (self.dim,)





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return len(self.event_shape       )





      @property
      def 	__UpperCamelCase(			self       ):
       '''simple docstring'''




       return 0.0





      def 	__UpperCamelCase(			self	,							A_       ):
       '''simple docstring'''




       return ParameterProjection(
           in_features=A_	,							args_dim=self.args_dim	,							domain_map=LambdaLayer(self.domain_map       )	,							)





      def 	__UpperCamelCase(			self	,							*A_       ):
       '''simple docstring'''




       raise NotImplementedError()





      @staticmethod
      def 	__UpperCamelCase(			A_       ):
       '''simple docstring'''




       return (x + torch.sqrt(torch.square(A_       ) + 4.0       )) / 2.0



class  A__       (	__snake_case      ):
      _UpperCAmelCase     :Dict[str, int]						   =       {"df": 1, "loc": 1, "scale": 1}
      _UpperCAmelCase     :type						   =       StudentT





      @classmethod
      def 	__UpperCamelCase(			cls	,							A_	,							A_	,							A_       ):
       '''simple docstring'''




       UpperCamelCase     : Optional[Any]	     =       cls.squareplus(A_       ).clamp_min(torch.finfo(scale.dtype       ).eps       )
       UpperCamelCase     : int	     =       2.0 + cls.squareplus(A_       )
       return df.squeeze(-1       ), loc.squeeze(-1       ), scale.squeeze(-1       )



class  A__       (	__snake_case      ):
      _UpperCAmelCase     :Dict[str, int]						   =       {"loc": 1, "scale": 1}
      _UpperCAmelCase     :type						   =       Normal





      @classmethod
      def 	__UpperCamelCase(			cls	,							A_	,							A_       ):
       '''simple docstring'''




       UpperCamelCase     : Dict	     =       cls.squareplus(A_       ).clamp_min(torch.finfo(scale.dtype       ).eps       )
       return loc.squeeze(-1       ), scale.squeeze(-1       )



class  A__       (	__snake_case      ):
      _UpperCAmelCase     :Dict[str, int]						   =       {"total_count": 1, "logits": 1}
      _UpperCAmelCase     :type						   =       NegativeBinomial





      @classmethod
      def 	__UpperCamelCase(			cls	,							A_	,							A_       ):
       '''simple docstring'''




       UpperCamelCase     : List[str]	     =       cls.squareplus(A_       )
       return total_count.squeeze(-1       ), logits.squeeze(-1       )





      def 	__UpperCamelCase(			self	,							A_       ):
       '''simple docstring'''




       UpperCamelCase     ,   UpperCamelCase     : Optional[int]	     =       distr_args
       if self.dim == 1:
        return self.distribution_class(total_count=A_	,							logits=A_       )
       else:
        return Independent(self.distribution_class(total_count=A_	,							logits=A_       )	,							1       )





      def 	__UpperCamelCase(			self	,							A_	,							A_ = None	,							A_ = None       ):
       '''simple docstring'''




       UpperCamelCase     ,   UpperCamelCase     : Any	     =       distr_args

       if scale is not None:
        # See scaling property of Gamma.
        logits += scale.log()

       return self._base_distribution((total_count, logits)       )
