'''simple docstring'''

from queue import PriorityQueue
from typing import Any

import numpy as np







def __UpperCamelCase(   _A					:				dict  ,			_A					:				str  ,			_A					:				set  ,			_A					:				set  ,			_A					:				dict  ,			_A					:				dict  ,			_A					:				PriorityQueue  ,			_A					:				dict  ,			_A					:				float | int  ,			):
   '''simple docstring'''
   for nxt, d in graph[v]:
      if nxt in visited_forward:
         continue
      UpperCAmelCase__							:				Union[str, Any]   				=  cst_fwd.get(_A  ,			np.inf    )
      UpperCAmelCase__							:				Optional[int]   				=  cst_fwd[v] + d
      if new_cost_f < old_cost_f:
         queue.put((new_cost_f, nxt)    )
         UpperCAmelCase__							:				Optional[Any]   				=  new_cost_f
         UpperCAmelCase__							:				int   				=  v
      if nxt in visited_backward:
         if cst_fwd[v] + d + cst_bwd[nxt] < shortest_distance:
            UpperCAmelCase__							:				Tuple   				=  cst_fwd[v] + d + cst_bwd[nxt]
   return shortest_distance







def __UpperCamelCase(   _A					:				str  ,			_A					:				str  ,			_A					:				dict  ,			_A					:				dict    ):
   '''simple docstring'''
   UpperCAmelCase__							:				Tuple   				=  -1

   UpperCAmelCase__							:				Union[str, Any]   				=  set()
   UpperCAmelCase__							:				Dict   				=  set()
   UpperCAmelCase__							:				Tuple   				=  {source: 0}
   UpperCAmelCase__							:				str   				=  {destination: 0}
   UpperCAmelCase__							:				int   				=  {source: None}
   UpperCAmelCase__							:				Tuple   				=  {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__							:				int   				=  queue_forward.get()
      visited_forward.add(_A    )

      UpperCAmelCase__							:				List[Any]   				=  queue_backward.get()
      visited_backward.add(_A    )

      UpperCAmelCase__							:				int   				=  pass_and_relaxation(
          _A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			)

      UpperCAmelCase__							:				Optional[Any]   				=  pass_and_relaxation(
          _A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			_A  ,			)

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

   if shortest_distance != np.inf:
      UpperCAmelCase__							:				Dict   				=  shortest_distance
   return shortest_path_distance


UpperCamelCase__     :	Tuple				=			{
    'B': [['C', 1]],
    'C': [['D', 1]],
    'D': [['F', 1]],
    'E': [['B', 1], ['G', 2]],
    'F': [],
    'G': [['F', 1]],
}
UpperCamelCase__     :	List[Any]				=			{
    '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()





'''simple docstring'''

import unittest

import numpy as np







def __UpperCamelCase(   _A					:				np.ndarray  ,			_A					:				np.ndarray  ,			_A					:				np.ndarray  ,			_A					:				np.ndarray | None = None  ,			):
   '''simple docstring'''
   UpperCAmelCase__							:				Any   				=  np.shape(_A    )
   UpperCAmelCase__							:				List[Any]   				=  np.shape(_A    )
   UpperCAmelCase__							:				Tuple   				=  np.shape(_A    )

   if shape_a[0] != shape_b[0]:
      UpperCAmelCase__							:				Any   				=  (
          '''Expected the same number of rows for A and B. '''
          F'''Instead found A of size {shape_a} and B of size {shape_b}'''
      )
      raise ValueError(_A    )

   if shape_b[1] != shape_c[1]:
      UpperCAmelCase__							:				List[str]   				=  (
          '''Expected the same number of columns for B and C. '''
          F'''Instead found B of size {shape_b} and C of size {shape_c}'''
      )
      raise ValueError(_A    )

   UpperCAmelCase__							:				Optional[Any]   				=  pseudo_inv
   if a_inv is None:
      try:
         UpperCAmelCase__							:				str   				=  np.linalg.inv(_A    )
      except np.linalg.LinAlgError:
         raise ValueError(
             '''Input matrix A is not invertible. Cannot compute Schur complement.'''    )

   return mat_c - mat_b.T @ a_inv @ mat_b


class 				_lowercase	(      unittest.TestCase   ):
      '''simple docstring'''


      def 		lowerCAmelCase__					(				self							)			->			None:



         '''simple docstring'''



         UpperCAmelCase__							:				int   				=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
         UpperCAmelCase__							:				List[Any]   				=  np.array([[0, 3], [3, 0], [2, 3]]							)
         UpperCAmelCase__							:				Optional[Any]   				=  np.array([[2, 1], [6, 3]]							)

         UpperCAmelCase__							:				Tuple   				=  schur_complement(lowerCamelCase_					,lowerCamelCase_					,lowerCamelCase_							)

         UpperCAmelCase__							:				Tuple   				=  np.block([[a, b], [b.T, c]]							)

         UpperCAmelCase__							:				int   				=  np.linalg.det(lowerCamelCase_							)
         UpperCAmelCase__							:				Optional[Any]   				=  np.linalg.det(lowerCamelCase_							)
         UpperCAmelCase__							:				Union[str, Any]   				=  np.linalg.det(lowerCamelCase_							)

         self.assertAlmostEqual(lowerCamelCase_					,det_a * det_s							)




      def 		lowerCAmelCase__					(				self							)			->			None:



         '''simple docstring'''



         UpperCAmelCase__							:				Optional[Any]   				=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
         UpperCAmelCase__							:				Tuple   				=  np.array([[0, 3], [3, 0], [2, 3]]							)
         UpperCAmelCase__							:				Tuple   				=  np.array([[2, 1], [6, 3]]							)

         with self.assertRaises(lowerCamelCase_							):
            schur_complement(lowerCamelCase_					,lowerCamelCase_					,lowerCamelCase_							)




      def 		lowerCAmelCase__					(				self							)			->			None:



         '''simple docstring'''



         UpperCAmelCase__							:				Dict   				=  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
         UpperCAmelCase__							:				str   				=  np.array([[0, 3], [3, 0], [2, 3]]							)
         UpperCAmelCase__							:				Any   				=  np.array([[2, 1, 3], [6, 3, 5]]							)

         with self.assertRaises(lowerCamelCase_							):
            schur_complement(lowerCamelCase_					,lowerCamelCase_					,lowerCamelCase_							)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  unittest.main()




