




def 					_a							(					lowerCAmelCase    ,						lowerCAmelCase    ,						lowerCAmelCase )->	bool:
		return not any(
		    neighbour == 1 and colored_vertices[i] == color
		    for i, neighbour in enumerate(lowerCAmelCase ) )


def 					_a							(					lowerCAmelCase    ,						lowerCAmelCase    ,						lowerCAmelCase    ,						lowerCAmelCase )->	bool:

		# Base Case
		if index == len(lowerCAmelCase ):
				return True

		# Recursive Step
		for i in range(lowerCAmelCase ):
				if valid_coloring(graph[index]    ,						lowerCAmelCase    ,						lowerCAmelCase ):
						# Color current vertex
						SCREAMING_SNAKE_CASE_					      =							i
						# Validate coloring
						if util_color(lowerCAmelCase    ,						lowerCAmelCase    ,						lowerCAmelCase    ,						index + 1 ):
								return True
						# Backtrack
						SCREAMING_SNAKE_CASE_					      =							-1
		return False




def 					_a							(					lowerCAmelCase    ,						lowerCAmelCase )->	list[int]:
		SCREAMING_SNAKE_CASE_					      =							[-1] * len(lowerCAmelCase )

		if util_color(lowerCAmelCase    ,						lowerCAmelCase    ,						lowerCAmelCase    ,						0 ):
				return colored_vertices

		return []





from dataclasses import dataclass
from typing import Tuple

import numpy as np
import torch



@dataclass
class    lowercase_		:
						lowerCAmelCase__			      =42  # [batch_size x 3]
						lowerCAmelCase__			      =42  # [batch_size x 3]
						lowerCAmelCase__			      =42  # [batch_size x 3]
						lowerCAmelCase__			      =42  # [batch_size x 3]
						lowerCAmelCase__			      =42
						lowerCAmelCase__			      =42
						lowerCAmelCase__			      =42
						lowerCAmelCase__			      =42
						lowerCAmelCase__			      =42



						def 	__a					(			self	:   Optional[Any]  ):

								"""simple docstring"""

								assert self.x.shape[0] == self.y.shape[0] == self.z.shape[0] == self.origin.shape[0]
								assert self.x.shape[1] == self.y.shape[1] == self.z.shape[1] == self.origin.shape[1] == 3
								assert len(self.x.shape  ) == len(self.y.shape  ) == len(self.z.shape  ) == len(self.origin.shape  ) == 2



						def 	__a					(			self	:   Union[str, Any]  ):

								"""simple docstring"""

								return torch.from_numpy(np.array([self.width, self.height]					,							dtype=np.floataa  )  )



						def 	__a					(			self	:   Optional[int]  ):

								"""simple docstring"""

								return torch.from_numpy(np.array([self.x_fov, self.y_fov]					,							dtype=np.floataa  )  )



						def 	__a					(			self	:   int  ):

								"""simple docstring"""

								SCREAMING_SNAKE_CASE_					      =							torch.arange(self.height * self.width  )
								SCREAMING_SNAKE_CASE_					      =							torch.stack(
								    [
								        pixel_indices % self.width,
								        torch.div(snake_case__					,							self.width					,							rounding_mode='trunc'  ),
								    ]					,							axis=1					,							)
								return coords



						@property
						def 	__a					(			self	:   List[Any]  ):

								"""simple docstring"""

								SCREAMING_SNAKE_CASE_      ,				*SCREAMING_SNAKE_CASE_					      =							self.shape
								SCREAMING_SNAKE_CASE_					      =							int(np.prod(snake_case__  )  )

								SCREAMING_SNAKE_CASE_					      =							self.get_image_coords()
								SCREAMING_SNAKE_CASE_					      =							torch.broadcast_to(coords.unsqueeze(0  )					,							[batch_size * inner_batch_size, *coords.shape]  )
								SCREAMING_SNAKE_CASE_					      =							self.get_camera_rays(snake_case__  )

								SCREAMING_SNAKE_CASE_					      =							rays.view(snake_case__					,							inner_batch_size * self.height * self.width					,							2					,							3  )

								return rays



						def 	__a					(			self	:   Optional[Any]					,							snake_case__	:   torch.Tensor  ):

								"""simple docstring"""

								SCREAMING_SNAKE_CASE_      ,				*SCREAMING_SNAKE_CASE_      ,				SCREAMING_SNAKE_CASE_					      =							coords.shape
								assert n_coords == 2
								assert batch_size == self.origin.shape[0]

								SCREAMING_SNAKE_CASE_					      =							coords.view(snake_case__					,							-1					,							2  )

								SCREAMING_SNAKE_CASE_					      =							self.resolution()
								SCREAMING_SNAKE_CASE_					      =							self.fov()

								SCREAMING_SNAKE_CASE_					      =							(flat.float() / (res - 1)) * 2 - 1
								SCREAMING_SNAKE_CASE_					      =							fracs * torch.tan(fov / 2  )

								SCREAMING_SNAKE_CASE_					      =							fracs.view(snake_case__					,							-1					,							2  )
								SCREAMING_SNAKE_CASE_					      =							(
								    self.z.view(snake_case__					,							1					,							3  )
								    + self.x.view(snake_case__					,							1					,							3  ) * fracs[:, :, :1]
								    + self.y.view(snake_case__					,							1					,							3  ) * fracs[:, :, 1:]
								)
								SCREAMING_SNAKE_CASE_					      =							directions / directions.norm(dim=-1					,							keepdim=snake_case__  )
								SCREAMING_SNAKE_CASE_					      =							torch.stack(
								    [
								        torch.broadcast_to(self.origin.view(snake_case__					,							1					,							3  )					,							[batch_size, directions.shape[1], 3]  ),
								        directions,
								    ]					,							dim=2					,							)
								return rays.view(snake_case__					,							*snake_case__					,							2					,							3  )






						def 	__a					(			self	:   Optional[int]					,							snake_case__	:   int					,							snake_case__	:   int  ):

								"""simple docstring"""

								assert width * self.height == height * self.width, "The aspect ratio should not change."
								return DifferentiableProjectiveCamera(
								    origin=self.origin					,							x=self.x					,							y=self.y					,							z=self.z					,							width=snake_case__					,							height=snake_case__					,							x_fov=self.x_fov					,							y_fov=self.y_fov					,							)


def 					_a							(					lowerCAmelCase )->	DifferentiableProjectiveCamera:
		SCREAMING_SNAKE_CASE_					      =							[]
		SCREAMING_SNAKE_CASE_					      =							[]
		SCREAMING_SNAKE_CASE_					      =							[]
		SCREAMING_SNAKE_CASE_					      =							[]
		for theta in np.linspace(0    ,						2 * np.pi    ,						num=20 ):
				SCREAMING_SNAKE_CASE_					      =							np.array([np.sin(lowerCAmelCase ), np.cos(lowerCAmelCase ), -0.5] )
				z /= np.sqrt(np.sum(z**2 ) )
				SCREAMING_SNAKE_CASE_					      =							-z * 4
				SCREAMING_SNAKE_CASE_					      =							np.array([np.cos(lowerCAmelCase ), -np.sin(lowerCAmelCase ), 0.0] )
				SCREAMING_SNAKE_CASE_					      =							np.cross(lowerCAmelCase    ,						lowerCAmelCase )
				origins.append(lowerCAmelCase )
				xs.append(lowerCAmelCase )
				ys.append(lowerCAmelCase )
				zs.append(lowerCAmelCase )
		return DifferentiableProjectiveCamera(
		    origin=torch.from_numpy(np.stack(lowerCAmelCase    ,						axis=0 ) ).float()    ,						x=torch.from_numpy(np.stack(lowerCAmelCase    ,						axis=0 ) ).float()    ,						y=torch.from_numpy(np.stack(lowerCAmelCase    ,						axis=0 ) ).float()    ,						z=torch.from_numpy(np.stack(lowerCAmelCase    ,						axis=0 ) ).float()    ,						width=lowerCAmelCase    ,						height=lowerCAmelCase    ,						x_fov=0.7    ,						y_fov=0.7    ,						shape=(1, len(lowerCAmelCase ))    ,						)