"""simple docstring"""



from dataclasses import dataclass
from typing import Optional, Tuple, Union

import flax
import jax.numpy as jnp
from jax import random

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput
from .scheduling_utils_flax import FlaxSchedulerMixin






@flax.struct.dataclass
class 				snake_case__     :
 # setable values
 _snake_case   :			Optional[int]           =		None
 _snake_case   :			Optional[jnp.ndarray]           =		None
 _snake_case   :			Optional[jnp.ndarray]           =		None  # sigma(t_i)


 @classmethod
 def 		a__ (				cls		):
        return cls()






@dataclass
class 				snake_case__     (   snake_case_   ):
 _snake_case   :			jnp.ndarray
 _snake_case   :			jnp.ndarray
 _snake_case   :			KarrasVeSchedulerState






class 				snake_case__     (   snake_case_,		snake_case_   ):


 @property
 def 		a__ (				self		):
        return True


 @register_to_config
 def __init__(				self  ,						lowerCamelCase = 0.02  ,						lowerCamelCase = 100  ,						lowerCamelCase = 1.007  ,						lowerCamelCase = 80  ,						lowerCamelCase = 0.05  ,						lowerCamelCase = 50  ,						):
        pass


 def 		a__ (				self		):
        return KarrasVeSchedulerState.create()


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase = ()		):
        __a     				=  jnp.arange(0  ,						lowerCamelCase		)[::-1].copy()
        __a     				=  [
            (
                self.config.sigma_max**2
                * (self.config.sigma_min**2 / self.config.sigma_max**2) ** (i / (num_inference_steps - 1))
            )
            for i in timesteps
        ]

        return state.replace(
            num_inference_steps=lowerCamelCase  ,						schedule=jnp.array(lowerCamelCase  ,						dtype=jnp.floataa		)  ,						timesteps=lowerCamelCase  ,						)


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						):
        if self.config.s_min <= sigma <= self.config.s_max:
               __a     				=  min(self.config.s_churn / state.num_inference_steps  ,						2**0.5 - 1		)
        else:
               __a     				=  0

        # sample eps ~ N(0, S_noise^2 * I)
        __a     				=  random.split(lowerCamelCase  ,						num=1		)
        __a     				=  self.config.s_noise * random.normal(key=lowerCamelCase  ,						shape=sample.shape		)
        __a     				=  sigma + gamma * sigma
        __a     				=  sample + ((sigma_hat**2 - sigma**2) ** 0.5 * eps)

        return sample_hat, sigma_hat


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase = True  ,						):
        __a     				=  sample_hat + sigma_hat * model_output
        __a     				=  (sample_hat - pred_original_sample) / sigma_hat
        __a     				=  sample_hat + (sigma_prev - sigma_hat) * derivative

        if not return_dict:
               return (sample_prev, derivative, state)

        return FlaxKarrasVeOutput(prev_sample=lowerCamelCase  ,						derivative=lowerCamelCase  ,						state=lowerCamelCase		)


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase = True  ,						):
        __a     				=  sample_prev + sigma_prev * model_output
        __a     				=  (sample_prev - pred_original_sample) / sigma_prev
        __a     				=  sample_hat + (sigma_prev - sigma_hat) * (0.5 * derivative + 0.5 * derivative_corr)

        if not return_dict:
               return (sample_prev, derivative, state)

        return FlaxKarrasVeOutput(prev_sample=lowerCamelCase  ,						derivative=lowerCamelCase  ,						state=lowerCamelCase		)






 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase		):
        raise NotImplementedError()

"""simple docstring"""



from __future__ import annotations



def 					_lowerCamelCase(					a ,  a ,  a ,  a ,  a ,  ):
       __a     				=  len(a							)

       # If row is equal to the size of the board it means there are a queen in each row in
       # the current board (possible_board)
       if row == n:
              # We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
              # this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
              boards.append([". " * i + "Q " + ". " * (n - 1 - i) for i in possible_board]							)
              return

       # We iterate each column in the row to find all possible results in each row
       for col in range(a							):
              # We apply that we learned previously. First we check that in the current board
              # (possible_board) there are not other same value because if there is it means
              # that there are a collision in vertical. Then we apply the two formulas we
              # learned before:
              #
              # 45º: y - x = b or 45: row - col = b
              # 135º: y + x = b or row + col = b.
              #
              # And we verify if the results of this two formulas not exist in their variables
              # respectively.  (diagonal_right_collisions, diagonal_left_collisions)
              #
              # If any or these are True it means there is a collision so we continue to the
              # next value in the for loop.
              if (
                  col in possible_board
                  or row - col in diagonal_right_collisions
                  or row + col in diagonal_left_collisions
              ):
                     continue

              # If it is False we call dfs function again and we update the inputs
              depth_first_search(
                  [*possible_board, col] ,  [*diagonal_right_collisions, row - col] ,  [*diagonal_left_collisions, row + col] ,  a ,  a ,  )



def 					_lowerCamelCase(					a							):
       __a     				=  []
       depth_first_search([] ,  [] ,  [] ,  a ,  a							)

       # Print all the boards
       for board in boards:
              for column in board:
                     print(a							)
              print(""							)

       print(len(a							) ,  "solutions were found."							)


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   n_queens_solution(4)
