




"""simple docstring"""

import unittest

import numpy as np






def 		_snake_case	(   _snake_case				:   np.ndarray							,    _snake_case				:   np.ndarray							,    _snake_case				:   np.ndarray							,    _snake_case				:   np.ndarray | None = None							,    )       ->   np.ndarray:






      '''simple docstring'''


      _A     =  np.shape(_snake_case				)
      _A     =  np.shape(_snake_case				)
      _A     =  np.shape(_snake_case				)

      if shape_a[0] != shape_b[0]:
            _A     =  (
                '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(_snake_case				)

      if shape_b[1] != shape_c[1]:
            _A     =  (
                '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(_snake_case				)

      _A     =  pseudo_inv
      if a_inv is None:
            try:
                  _A     =  np.linalg.inv(_snake_case				)
            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			:			Optional[int]						):
       _A     =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]						)
       _A     =  np.array([[0, 3], [3, 0], [2, 3]]						)
       _A     =  np.array([[2, 1], [6, 3]]						)

       _A     =  schur_complement(_UpperCAmelCase       ,				_UpperCAmelCase       ,				_UpperCAmelCase						)

       _A     =  np.block([[a, b], [b.T, c]]						)

       _A     =  np.linalg.det(_UpperCAmelCase						)
       _A     =  np.linalg.det(_UpperCAmelCase						)
       _A     =  np.linalg.det(_UpperCAmelCase						)

       self.assertAlmostEqual(_UpperCAmelCase       ,				det_a * det_s						)

 def      lowerCAmelCase_				(					self			:			str						):
       _A     =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]						)
       _A     =  np.array([[0, 3], [3, 0], [2, 3]]						)
       _A     =  np.array([[2, 1], [6, 3]]						)

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



 def      lowerCAmelCase_				(					self			:			List[Any]						):
       _A     =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]						)
       _A     =  np.array([[0, 3], [3, 0], [2, 3]]						)
       _A     =  np.array([[2, 1, 3], [6, 3, 5]]						)

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


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       unittest.main()


'''simple docstring'''




# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.






def 							snake_case__				(     _A:   Dict					)       ->      str:

		'''simple docstring'''


		lowerCAmelCase     =    [False] * len(_A					)
		lowerCAmelCase     =    [-1] * len(_A					)

		def dfs(_A:   Dict ,  _A:   Tuple					):
				lowerCAmelCase     =    True
				lowerCAmelCase     =    c
				for u in graph[v]:
						if not visited[u]:
								dfs(_A ,  1 - c					)

		for i in range(len(_A					)					):
				if not visited[i]:
						dfs(_A ,  0					)

		for i in range(len(_A					)					):
				for j in graph[i]:
						if color[i] == color[j]:
								return False

		return True


# Adjacency list of graph
__lowercase        =    {0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2], 4: []}




print(check_bipartite_dfs(graph))




