



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






def  UpperCAmelCase__    (    lowercase__   ) ->      str:
						return np.dot(snake_case_						,       snake_case_   )
class 			_lowerCAmelCase   :






			"""simple docstring"""





			def __init__(				self	:       Union[str, Any]			,     *,
			    lowercase	:       Tuple = np.inf			,     lowercase	:       int = "linear"			,     lowercase	:       List[Any] = 0.0			,     )						->					None:
									"""simple docstring"""







									__lowercase					=				regularization
									__lowercase					=				gamma
									if kernel == "linear":
															__lowercase					=				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"""   )
															__lowercase					=				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:
															__lowercase					=				F"Unknown kernel: {kernel}"
															raise ValueError(_a   )





			def 		snake_case__			(				self	:       Any			,     lowercase	:       Optional[int]			,     lowercase	:       Optional[Any]   )						->					float:
									"""simple docstring"""







									return np.dot(_a			,     _a   )





			def 		snake_case__			(				self	:       str			,     lowercase	:       Union[str, Any]			,     lowercase	:       Optional[Any]   )						->					float:
									"""simple docstring"""







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





			def 		snake_case__			(				self	:       Any			,     lowercase	:       Union[str, Any]			,     lowercase	:       int   )						->					None:
									"""simple docstring"""







									__lowercase					=				observations
									__lowercase					=				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

									(__lowercase     )					=				np.shape(_a   )

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

									__lowercase					=				LinearConstraint(_a			,     0			,     0   )
									__lowercase					=				Bounds(0			,     self.regularization   )

									__lowercase					=				minimize(
									    _a			,     np.ones(_a   )			,     bounds=_a			,     constraints=[ly_contraint]   ).x
									__lowercase					=				l_star

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







			def 		snake_case__			(				self	:       Any			,     lowercase	:       str   )						->					int:
									"""simple docstring"""







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


if __name__ == "__main__":
			import doctest

			doctest.testmod()










import random






def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__ = False   ) ->      dict:
						__lowercase					=				{i: [] for i in range(lowercase__   )}

						# if probability is greater or equal than 1, then generate a complete graph
						if probability >= 1:
												return complete_graph(lowercase__   )
						# if probability is lower or equal than 0, then return a graph without edges
						if probability <= 0:
												return graph

						# for each couple of nodes, add an edge from u to v
						# if the number randomly generated is greater than probability probability
						for i in range(lowercase__   ):
												for j in range(i + 1						,       lowercase__   ):
																		if random.random() < probability:
																								graph[i].append(lowercase__   )
																								if not directed:
																														# if the graph is undirected, add an edge in from j to i, either
																														graph[j].append(lowercase__   )
						return graph






def  UpperCAmelCase__    (    lowercase__   ) ->      dict:
						return {
						    i: [j for j in range(lowercase__   ) if i != j] for i in range(lowercase__   )
						}


if __name__ == "__main__":
			import doctest

			doctest.testmod()





