
"""simple docstring"""






from __future__ import annotations
def 					UpperCamelCase		( UpperCAmelCase      ,					UpperCAmelCase      ,					UpperCAmelCase      ,					UpperCAmelCase      ,					UpperCAmelCase      ,					)		->None:

   """simple docstring"""



   a_							=			len(__lowercase			)

   # 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(__lowercase			):
      # 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]      ,					__lowercase      ,					__lowercase      ,					)
def 					UpperCamelCase		( UpperCAmelCase			)		->None:

   """simple docstring"""



   a_							=			[]
   depth_first_search([]      ,					[]      ,					[]      ,					__lowercase      ,					__lowercase			)

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

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


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      n_queens_solution(4)

from __future__ import annotations

import time
from collections.abc import Sequence
from random import randint

from matplotlib import pyplot as plt






def        a_		(       __lowercase   :	Sequence[float]       ,					__lowercase   :	int       ,					__lowercase   :	int ) ->  tuple[int | None, int | None, float]:
			if not arr:
						return None, None, 0
			if low == high:
						return low, high, arr[low]

			_snake_case		=			(low + high) // 2
			_snake_case	,					_snake_case	,					_snake_case		=			max_subarray(__lowercase       ,					__lowercase       ,					__lowercase )
			_snake_case	,					_snake_case	,					_snake_case		=			max_subarray(__lowercase       ,					mid + 1       ,					__lowercase )
			_snake_case	,					_snake_case	,					_snake_case		=			max_cross_sum(__lowercase       ,					__lowercase       ,					__lowercase       ,					__lowercase )
			if left_sum >= right_sum and left_sum >= cross_sum:
						return left_low, left_high, left_sum
			elif right_sum >= left_sum and right_sum >= cross_sum:
						return right_low, right_high, right_sum
			return cross_left, cross_right, cross_sum






def        a_		(       __lowercase   :	Sequence[float]       ,					__lowercase   :	int       ,					__lowercase   :	int       ,					__lowercase   :	int ) ->  tuple[int, int, float]:
			_snake_case	,					_snake_case		=			float('-inf' ), -1
			_snake_case	,					_snake_case		=			float('-inf' ), -1

			_snake_case		=			0
			for i in range(__lowercase       ,					low - 1       ,					-1 ):
						summ += arr[i]
						if summ > left_sum:
									_snake_case		=			summ
									_snake_case		=			i

			_snake_case		=			0
			for i in range(mid + 1       ,					high + 1 ):
						summ += arr[i]
						if summ > right_sum:
									_snake_case		=			summ
									_snake_case		=			i

			return max_left, max_right, (left_sum + right_sum)






def        a_		(       __lowercase   :	int ) ->  float:
			_snake_case		=			[randint(1       ,					__lowercase ) for _ in range(__lowercase )]
			_snake_case		=			time.time()
			max_subarray(__lowercase       ,					0       ,					input_size - 1 )
			_snake_case		=			time.time()
			return end - start






def        a_		(       ) ->  None:
			_snake_case		=			[10, 100, 1_000, 10_000, 50_000, 100_000, 200_000, 300_000, 400_000, 500_000]
			_snake_case		=			[time_max_subarray(__lowercase ) for input_size in input_sizes]
			print('No of Inputs\t\tTime Taken' )
			for input_size, runtime in zip(__lowercase       ,					__lowercase ):
						print(__lowercase       ,					'\t\t'       ,					__lowercase )
			plt.plot(__lowercase       ,					__lowercase )
			plt.xlabel('Number of Inputs' )
			plt.ylabel('Time taken in seconds' )
			plt.show()


if __name__ == "__main__":
			from doctest import testmod

			testmod()