'''simple docstring'''


from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve


def    _lowerCamelCase       ( lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				np.ndarray  ):



     """simple docstring"""





     UpperCAmelCase_				:  int  = np.stack((horizontal_flow, vertical_flow)      ,	2  )

     # Create a grid of all pixel coordinates and subtract the flow to get the
     # target pixels coordinates
     UpperCAmelCase_				:  str  = np.stack(
         np.meshgrid(np.arange(0      ,	image.shape[1]  )      ,	np.arange(0      ,	image.shape[0]  )  )      ,	2  )
     UpperCAmelCase_				:  Optional[int]  = np.round(grid - flow  ).astype(np.intaa  )

     # Find the locations outside of the original image
     UpperCAmelCase_				:  Any  = (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]  ))
     UpperCAmelCase_				:  Tuple  = 0

     UpperCAmelCase_				:  List[str]  = image[grid[:, :, 1], grid[:, :, 0]]

     # Set pixels at invalid locations to 0
     UpperCAmelCase_				:  Union[str, Any]  = 0

     return warped


def    _lowerCamelCase       ( lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				SupportsIndex      ,	lowerCamelCase_   :				float | None = None      ,	):



     """simple docstring"""





     if alpha is None:
          UpperCAmelCase_				:  Any  = 0.1

     # Initialize flow
     UpperCAmelCase_				:  Tuple  = np.zeros_like(lowerCamelCase_  )
     UpperCAmelCase_				:  Union[str, Any]  = np.zeros_like(lowerCamelCase_  )

     # Prepare kernels for the calculation of the derivatives and the average velocity
     UpperCAmelCase_				:  str  = np.array([[-1, 1], [-1, 1]]  ) * 0.25
     UpperCAmelCase_				:  int  = np.array([[-1, -1], [1, 1]]  ) * 0.25
     UpperCAmelCase_				:  int  = np.array([[1, 1], [1, 1]]  ) * 0.25
     UpperCAmelCase_				:  int  = np.array(
         [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]  )

     # Iteratively refine the flow
     for _ in range(lowerCamelCase_  ):
          UpperCAmelCase_				:  Dict  = warp(lowerCamelCase_      ,	lowerCamelCase_      ,	lowerCamelCase_  )
          UpperCAmelCase_				:  List[str]  = convolve(lowerCamelCase_      ,	lowerCamelCase_  ) + convolve(lowerCamelCase_      ,	lowerCamelCase_  )
          UpperCAmelCase_				:  str  = convolve(lowerCamelCase_      ,	lowerCamelCase_  ) + convolve(lowerCamelCase_      ,	lowerCamelCase_  )
          UpperCAmelCase_				:  int  = convolve(lowerCamelCase_      ,	lowerCamelCase_  ) + convolve(lowerCamelCase_      ,	-kernel_t  )

          UpperCAmelCase_				:  Optional[Any]  = convolve(lowerCamelCase_      ,	lowerCamelCase_  )
          UpperCAmelCase_				:  List[str]  = convolve(lowerCamelCase_      ,	lowerCamelCase_  )

          # This updates the flow as proposed in the paper (Step 12)
          UpperCAmelCase_				:  List[Any]  = (
              derivative_x * avg_horizontal_velocity
              + derivative_y * avg_vertical_velocity
              + derivative_t
          )
          UpperCAmelCase_				:  Optional[int]  = update / (alpha**2 + derivative_x**2 + derivative_y**2)

          UpperCAmelCase_				:  str  = avg_horizontal_velocity - derivative_x * update
          UpperCAmelCase_				:  Optional[int]  = avg_vertical_velocity - derivative_y * update

     return horizontal_flow, vertical_flow


if __name__ == "__main__":
 import doctest

 doctest.testmod()











# 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.
from queue import Queue


def 						_UpperCAmelCase							( snake_case     ):





					"""simple docstring"""

					_lowerCAmelCase         =       Queue()
					_lowerCAmelCase         =       [False] * len(snake_case     )
					_lowerCAmelCase         =       [-1] * len(snake_case     )

					def bfs():
										while not queue.empty():
															_lowerCAmelCase         =       queue.get()
															_lowerCAmelCase         =       True

															for neighbour in graph[u]:
																				if neighbour == u:
																									return False

																				if color[neighbour] == -1:
																									_lowerCAmelCase         =       1 - color[u]
																									queue.put(snake_case     )

																				elif color[neighbour] == color[u]:
																									return False

										return True

					for i in range(len(snake_case     )     ):
										if not visited[i]:
															queue.put(snake_case     )
															_lowerCAmelCase         =       0
															if bfs() is False:
																				return False

					return True


if __name__ == "__main__":
	# Adjacency List of graph
	print(check_bipartite({0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2]}))



