

import numpy as np
from numpy import ndarray
from scipy.optimize import Bounds, LinearConstraint, minimize



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							ndarray      )			->      float:




				'''simple docstring'''





				return np.dot(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )

class 					a__						:
			"""simple docstring"""







			def __init__(    self      ,      *,
			    lowercase = np.inf      ,      lowercase = "linear"      ,      lowercase = 0.0      ,      )  ->						None:
							'''simple docstring'''






							A__        =			regularization
							A__        =			gamma
							if kernel == "linear":
											A__        =			self.__linear
							elif kernel == "rbf":
											if self.gamma == 0:
															raise ValueError("rbf kernel requires gamma"       )
											if not isinstance(self.gamma      ,      (float, int)       ):
															raise ValueError("gamma must be float or int"       )
											if not self.gamma > 0:
															raise ValueError("gamma must be > 0"       )
											A__        =			self.__rbf
											# in the future, there could be a default value like in sklearn
											# sklear: def_gamma = 1/(n_features * X.var()) (wiki)
											# previously it was 1/(n_features)
							else:
											A__        =			F'Unknown kernel: {kernel}'
											raise ValueError(lowercase       )


			def      UpperCamelCase		(    self      ,      lowercase      ,      lowercase       )  ->						float:
							'''simple docstring'''






							return np.dot(lowercase      ,      lowercase       )


			def      UpperCamelCase		(    self      ,      lowercase      ,      lowercase       )  ->						float:
							'''simple docstring'''






							return np.exp(-(self.gamma * norm_squared(vectora - vectora       ))       )


			def      UpperCamelCase		(    self      ,      lowercase      ,      lowercase       )  ->						None:
							'''simple docstring'''






							A__        =			observations
							A__        =			classes

							# using Wolfe's Dual to calculate w.
							# Primal problem: minimize 1/2*norm_squared(w)
							#   constraint: yn(w . xn + b) >= 1
							#
							# With l a vector
							# Dual problem: maximize sum_n(ln) -
							#       1/2 * sum_n(sum_m(ln*lm*yn*ym*xn . xm))
							#   constraint: self.C >= ln >= 0
							#           and sum_n(ln*yn) = 0
							# Then we get w using w = sum_n(ln*yn*xn)
							# At the end we can get b ~= mean(yn - w . xn)
							#
							# Since we use kernels, we only need l_star to calculate b
							# and to classify observations

							((A__)    ,				)        =			np.shape(lowercase       )

							def to_minimize(lowercase       ) -> float:
											A__        =			0
											((A__)    ,				)        =			np.shape(lowercase       )
											for i in range(lowercase       ):
															for j in range(lowercase       ):
																			s += (
																			    candidate[i]
																			    * candidate[j]
																			    * classes[i]
																			    * classes[j]
																			    * self.kernel(observations[i]      ,      observations[j]       )
																			)
											return 1 / 2 * s - sum(lowercase       )

							A__        =			LinearConstraint(lowercase      ,      0      ,      0       )
							A__        =			Bounds(0      ,      self.regularization       )

							A__        =			minimize(
							    lowercase      ,      np.ones(lowercase       )      ,      bounds=lowercase      ,      constraints=[ly_contraint]       ).x
							A__        =			l_star

							# calculating mean offset of separation plane to points
							A__        =			0
							for i in range(lowercase       ):
											for j in range(lowercase       ):
															s += classes[i] - classes[i] * self.optimum[i] * self.kernel(
															    observations[i]      ,      observations[j]       )
							A__        =			s / n


			def      UpperCamelCase		(    self      ,      lowercase       )  ->						int:
							'''simple docstring'''






							A__        =			sum(
							    self.optimum[n]
							    * self.classes[n]
							    * self.kernel(self.observations[n]      ,      lowercase       )
							    for n in range(len(self.classes       )       )       )
							return 1 if s + self.offset >= 0 else -1


if __name__ == "__main__":
	import doctest

	doctest.testmod()




from __future__ import annotations



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[list[str]]	,       SCREAMING_SNAKE_CASE_:							int	,       )			->      None:




				'''simple docstring'''





				A__        =			len(SCREAMING_SNAKE_CASE_      )

				# 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(SCREAMING_SNAKE_CASE_      ):
								# 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]	,       SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_	,       )



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							int      )			->      None:




				'''simple docstring'''





				A__        =			[]
				depth_first_search([]	,       []	,       []	,       SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )

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

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	n_queens_solution(4)

