'''simple docstring'''




import heapq

def 			UpperCamelCase_							(      _UpperCAmelCase						:   List[Any]    ,			_UpperCAmelCase						:   Optional[Any]    ,			_UpperCAmelCase						:   Union[str, Any]					)					->					Tuple:


       """simple docstring"""







       _UpperCAmelCase  :						str							   =					[(0, start)]  # cost from start node,end node
       _UpperCAmelCase  :						List[str]							   =					set()
       while heap:
              ((_UpperCAmelCase)   ,						(_UpperCAmelCase))  :						Any							   =					heapq.heappop(_UpperCAmelCase					)
              if u in visited:
                     continue
              visited.add(_UpperCAmelCase					)
              if u == end:
                     return cost
              for v, c in graph[u]:
                     if v in visited:
                            continue
                     _UpperCAmelCase  :						str							   =					cost + c
                     heapq.heappush(_UpperCAmelCase    ,			(next_item, v)					)
       return -1


__SCREAMING_SNAKE_CASE :  Tuple  =						{
    """A""": [["""B""", 2], ["""C""", 5]],
    """B""": [["""A""", 2], ["""D""", 3], ["""E""", 1], ["""F""", 1]],
    """C""": [["""A""", 5], ["""F""", 3]],
    """D""": [["""B""", 3]],
    """E""": [["""B""", 4], ["""F""", 3]],
    """F""": [["""C""", 3], ["""E""", 3]],
}
__SCREAMING_SNAKE_CASE :  Union[str, Any]  =						{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""F""", 3]],
    """F""": [],
}
__SCREAMING_SNAKE_CASE :  str  =						{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""G""", 2]],
    """F""": [],
    """G""": [["""F""", 1]],
}

__SCREAMING_SNAKE_CASE :  Tuple  =						dijkstra(G, """E""", """C""")
print(short_distance)  # E -- 3 --> F -- 3 --> C == 6

__SCREAMING_SNAKE_CASE :  Tuple  =						dijkstra(Ga, """E""", """F""")
print(short_distance)  # E -- 3 --> F == 3

__SCREAMING_SNAKE_CASE :  Optional[int]  =						dijkstra(Ga, """E""", """F""")
print(short_distance)  # E -- 2 --> G -- 1 --> F == 3

if __name__ == "__main__":
  import doctest

  doctest.testmod()


'''simple docstring'''




from typing import Any

import numpy as np

class 			lowerCamelCase_			:
      '''simple docstring'''

      def __init__( self		:       Optional[int]		,      A		:       np.ndarray		,      A		:       int ):
             # Check if RHS is negative
             if np.any(tableau[:, -1]		,      where=tableau[:, -1] < 0 ):
                    raise ValueError("RHS must be > 0" )

             _UpperCAmelCase  :						Union[str, Any]							   =					tableau
             _UpperCAmelCase   ,						_UpperCAmelCase  :						str							   =					tableau.shape

             # Number of decision variables x1, x2, x3...
             _UpperCAmelCase  :						Any							   =					n_vars

             # Number of artificial variables to be minimised
             _UpperCAmelCase  :						List[str]							   =					len(np.where(tableau[self.n_vars : -1] == -1 )[0] )

             # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
             _UpperCAmelCase  :						List[Any]							   =					(self.n_art_vars > 0) + 1

             # Number of slack variables added to make inequalities into equalities
             _UpperCAmelCase  :						int							   =					self.n_rows - self.n_stages

             # Objectives for each stage
             _UpperCAmelCase  :						Optional[Any]							   =					["max"]

             # In two stage simplex, first minimise then maximise
             if self.n_art_vars:
                    self.objectives.append("min" )

             _UpperCAmelCase  :						Any							   =					[""]

             # Index of current pivot row and column
             _UpperCAmelCase  :						Optional[Any]							   =					None
             _UpperCAmelCase  :						Dict							   =					None

             # Does objective row only contain (non)-negative values?
             _UpperCAmelCase  :						List[str]							   =					False




      @staticmethod
      def   _A			( *A		:       int ):
             if len(A ) != 3:
                    raise ValueError("Must provide n_vars, n_slack, and n_art_vars" )

             if not all(x >= 0 and isinstance(A		,      A ) for x in args ):
                    raise ValueError("All arguments must be non-negative integers" )

             # decision | slack | artificial
             _UpperCAmelCase  :						Optional[Any]							   =					["x", "s", "a"]
             _UpperCAmelCase  :						int							   =					[]
             for i in range(3 ):
                    for j in range(args[i] ):
                           titles.append(string_starts[i] + str(j + 1 ) )
             titles.append("RHS" )
             return titles




      def   _A			( self		:       List[str]		,      A		:       np.ndarray ):
             _UpperCAmelCase  :						str							   =					self.objectives[-1]

             # Find entries of highest magnitude in objective rows
             _UpperCAmelCase  :						List[str]							   =					(objective == "min") - (objective == "max")
             _UpperCAmelCase  :						Optional[Any]							   =					np.argmax(sign * tableau[0, : self.n_vars] )

             # Choice is only valid if below 0 for maximise, and above for minimise
             if sign * self.tableau[0, col_idx] <= 0:
                    _UpperCAmelCase  :						Dict							   =					True
                    return 0, 0

             # Pivot row is chosen as having the lowest quotient when elements of
             # the pivot column divide the right-hand side

             # Slice excluding the objective rows
             _UpperCAmelCase  :						Any							   =					slice(self.n_stages		,      self.n_rows )

             # RHS
             _UpperCAmelCase  :						Optional[Any]							   =					tableau[s, -1]

             # Elements of pivot column within slice
             _UpperCAmelCase  :						Optional[Any]							   =					tableau[s, col_idx]

             # Array filled with nans
             _UpperCAmelCase  :						List[Any]							   =					np.full(self.n_rows - self.n_stages		,      np.nan )

             # If element in pivot column is greater than zeron_stages, return
             # quotient or nan otherwise
             _UpperCAmelCase  :						str							   =					np.divide(A		,      A		,      out=A		,      where=divisor > 0 )

             # Arg of minimum quotient excluding the nan values. n_stages is added
             # to compensate for earlier exclusion of objective columns
             _UpperCAmelCase  :						Optional[Any]							   =					np.nanargmin(A ) + self.n_stages
             return row_idx, col_idx




      def   _A			( self		:       List[Any]		,      A		:       np.ndarray		,      A		:       int		,      A		:       int ):
             _UpperCAmelCase  :						Union[str, Any]							   =					tableau[row_idx].copy()

             _UpperCAmelCase  :						Optional[Any]							   =					piv_row[col_idx]

             # Entry becomes 1
             piv_row *= 1 / piv_val

             # Variable in pivot column becomes basic, ie the only non-zero entry
             for idx, coeff in enumerate(tableau[:, col_idx] ):
                    tableau[idx] += -coeff * piv_row
             _UpperCAmelCase  :						Optional[Any]							   =					piv_row
             return tableau




      def   _A			( self		:       Optional[Any]		,      A		:       np.ndarray ):
             self.objectives.pop()

             if not self.objectives:
                    return tableau

             # Slice containing ids for artificial columns
             _UpperCAmelCase  :						Union[str, Any]							   =					slice(-self.n_art_vars - 1		,      -1 )

             # Delete the artificial variable columns
             _UpperCAmelCase  :						List[Any]							   =					np.delete(A		,      A		,      axis=1 )

             # Delete the objective row of the first stage
             _UpperCAmelCase  :						Union[str, Any]							   =					np.delete(A		,      0		,      axis=0 )

             _UpperCAmelCase  :						Any							   =					1
             self.n_rows -= 1
             _UpperCAmelCase  :						Dict							   =					0
             _UpperCAmelCase  :						List[str]							   =					False
             return tableau




      def   _A			( self		:       Optional[Any] ):
             # Stop simplex algorithm from cycling.
             for _ in range(100 ):
                    # Completion of each stage removes an objective. If both stages
                    # are complete, then no objectives are left
                    if not self.objectives:
                           _UpperCAmelCase  :						Any							   =					self.generate_col_titles(
                               self.n_vars		,      self.n_slack		,      self.n_art_vars )

                           # Find the values of each variable at optimal solution
                           return self.interpret_tableau(self.tableau		,      self.col_titles )

                    _UpperCAmelCase   ,						_UpperCAmelCase  :						List[str]							   =					self.find_pivot(self.tableau )

                    # If there are no more negative values in objective row
                    if self.stop_iter:
                           # Delete artificial variable columns and rows. Update attributes
                           _UpperCAmelCase  :						int							   =					self.change_stage(self.tableau )
                    else:
                           _UpperCAmelCase  :						Optional[Any]							   =					self.pivot(self.tableau		,      A		,      A )
             return {}





      def   _A			( self		:       List[Any]		,      A		:       np.ndarray		,      A		:       list[str] ):
             _UpperCAmelCase  :						str							   =					{"P": abs(tableau[0, -1] )}

             for i in range(self.n_vars ):
                    # Gives ids of nonzero entries in the ith column
                    _UpperCAmelCase  :						Dict							   =					np.nonzero(tableau[:, i] )
                    _UpperCAmelCase  :						Dict							   =					len(nonzero[0] )

                    # First entry in the nonzero ids
                    _UpperCAmelCase  :						str							   =					nonzero[0][0]
                    _UpperCAmelCase  :						str							   =					tableau[nonzero_rowidx, i]

                    # If there is only one nonzero value in column, which is one
                    if n_nonzero == nonzero_val == 1:
                           _UpperCAmelCase  :						Optional[Any]							   =					tableau[nonzero_rowidx, -1]
                           _UpperCAmelCase  :						Any							   =					rhs_val

        # Check for basic variables
             for title in col_titles:
                    # Don't add RHS or slack variables to output dict
                    if title[0] not in "R-s-a":
                           output_dict.setdefault(A		,      0 )
             return output_dict


if __name__ == "__main__":
  import doctest

  doctest.testmod()

