





from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload






class 					lowercase__ :

   '''simple docstring'''


   def __init__(					self,	__magic_name__ = None							)				->   None:






       """simple docstring"""
       if components is None:
           UpperCamelCase__							:       Optional[Any]      =   []
       UpperCamelCase__							:       List[str]      =   list(__magic_name__							)




   def __len__(					self							)				->   int:






       """simple docstring"""
       return len(self.__components							)




   def __str__(					self							)				->   str:






       """simple docstring"""
       return "(" + ",".join(map(__magic_name__,	self.__components							)							) + ")"




   def __add__(					self,	__magic_name__							)				->   Vector:






       """simple docstring"""
       UpperCamelCase__							:       Optional[int]      =   len(self							)
       if size == len(__magic_name__							):
           UpperCamelCase__							:       Optional[Any]      =   [self.__components[i] + other.component(__magic_name__							) for i in range(__magic_name__							)]
           return Vector(__magic_name__							)
       else:
           raise Exception('''must have the same size'''							)




   def __sub__(					self,	__magic_name__							)				->   Vector:






       """simple docstring"""
       UpperCamelCase__							:       str      =   len(self							)
       if size == len(__magic_name__							):
           UpperCamelCase__							:       Optional[int]      =   [self.__components[i] - other.component(__magic_name__							) for i in range(__magic_name__							)]
           return Vector(__magic_name__							)
       else:  # error case
           raise Exception('''must have the same size'''							)




   @overload
   def __mul__(					self,	__magic_name__							)				->   Vector:






       """simple docstring"""
       ...




   @overload
   def __mul__(					self,	__magic_name__							)				->   float:






       """simple docstring"""
       ...




   def __mul__(					self,	__magic_name__							)				->   float | Vector:






       """simple docstring"""
       if isinstance(__magic_name__,	(float, int)							):
           UpperCamelCase__							:       str      =   [c * other for c in self.__components]
           return Vector(__magic_name__							)
       elif isinstance(__magic_name__,	__magic_name__							) and len(self							) == len(__magic_name__							):
           UpperCamelCase__							:       List[Any]      =   len(self							)
           UpperCamelCase__							:       Optional[int]      =   [self.__components[i] * other.component(__magic_name__							) for i in range(__magic_name__							)]
           return sum(__magic_name__							)
       else:  # error case
           raise Exception('''invalid operand!'''							)




   def    UpperCamelCase__						(					self							)				->   Vector:






       """simple docstring"""
       return Vector(self.__components							)




   def    UpperCamelCase__						(					self,	__magic_name__							)				->   float:






       """simple docstring"""
       if isinstance(__magic_name__,	__magic_name__							) and -len(self.__components							) <= i < len(self.__components							):
           return self.__components[i]
       else:
           raise Exception('''index out of range'''							)




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__							)				->   None:






       """simple docstring"""
       assert -len(self.__components							) <= pos < len(self.__components							)
       UpperCamelCase__							:       int      =   value




   def    UpperCamelCase__						(					self							)				->   float:






       """simple docstring"""
       if len(self.__components							) == 0:
           raise Exception('''Vector is empty'''							)
       UpperCamelCase__							:       Tuple      =   [c**2 for c in self.__components]
       return math.sqrt(sum(__magic_name__							)							)




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__ = False							)				->   float:






       """simple docstring"""
       UpperCamelCase__							:       Tuple      =   self * other
       UpperCamelCase__							:       Union[str, Any]      =   self.euclidean_length() * other.euclidean_length()
       if deg:
           return math.degrees(math.acos(num / den							)							)
       else:
           return math.acos(num / den							)




def 	lowerCAmelCase_					(						__UpperCAmelCase:    int   )   ->			Vector:
    assert isinstance(__UpperCAmelCase	,     __UpperCAmelCase   )
    return Vector([0] * dimension   )




def 	lowerCAmelCase_					(						__UpperCAmelCase:    int	,     __UpperCAmelCase:    int   )   ->			Vector:
    assert isinstance(__UpperCAmelCase	,     __UpperCAmelCase   ) and (isinstance(__UpperCAmelCase	,     __UpperCAmelCase   ))
    UpperCamelCase__							:       Tuple      =   [0] * dimension
    UpperCamelCase__							:       Tuple      =   1
    return Vector(__UpperCAmelCase   )




def 	lowerCAmelCase_					(						__UpperCAmelCase:    float	,     __UpperCAmelCase:    Vector	,     __UpperCAmelCase:    Vector   )   ->			Vector:
    assert (
        isinstance(__UpperCAmelCase	,     __UpperCAmelCase   )
        and isinstance(__UpperCAmelCase	,     __UpperCAmelCase   )
        and (isinstance(__UpperCAmelCase	,     (int, float)   ))
    )
    return x * scalar + y




def 	lowerCAmelCase_					(						__UpperCAmelCase:    int	,     __UpperCAmelCase:    int	,     __UpperCAmelCase:    int   )   ->			Vector:
    random.seed(__UpperCAmelCase   )
    UpperCamelCase__							:       List[Any]      =   [random.randint(__UpperCAmelCase	,     __UpperCAmelCase   ) for _ in range(__UpperCAmelCase   )]
    return Vector(__UpperCAmelCase   )






class 					lowercase__ :

   '''simple docstring'''


   def __init__(					self,	__magic_name__,	__magic_name__,	__magic_name__							)				->   None:






       """simple docstring"""
       UpperCamelCase__							:       Any      =   matrix
       UpperCamelCase__							:       List[Any]      =   w
       UpperCamelCase__							:       Union[str, Any]      =   h




   def __str__(					self							)				->   str:






       """simple docstring"""
       UpperCamelCase__							:       int      =   ''''''
       for i in range(self.__height							):
           ans += "|"
           for j in range(self.__width							):
               if j < self.__width - 1:
                   ans += str(self.__matrix[i][j]							) + ","
               else:
                   ans += str(self.__matrix[i][j]							) + "|\n"
       return ans




   def __add__(					self,	__magic_name__							)				->   Matrix:






       """simple docstring"""
       if self.__width == other.width() and self.__height == other.height():
           UpperCamelCase__							:       Tuple      =   []
           for i in range(self.__height							):
               UpperCamelCase__							:       Any      =   [
                   self.__matrix[i][j] + other.component(__magic_name__,	__magic_name__							)
                   for j in range(self.__width							)
               ]
               matrix.append(__magic_name__							)
           return Matrix(__magic_name__,	self.__width,	self.__height							)
       else:
           raise Exception('''matrix must have the same dimension!'''							)




   def __sub__(					self,	__magic_name__							)				->   Matrix:






       """simple docstring"""
       if self.__width == other.width() and self.__height == other.height():
           UpperCamelCase__							:       List[Any]      =   []
           for i in range(self.__height							):
               UpperCamelCase__							:       Optional[Any]      =   [
                   self.__matrix[i][j] - other.component(__magic_name__,	__magic_name__							)
                   for j in range(self.__width							)
               ]
               matrix.append(__magic_name__							)
           return Matrix(__magic_name__,	self.__width,	self.__height							)
       else:
           raise Exception('''matrices must have the same dimension!'''							)




   @overload
   def __mul__(					self,	__magic_name__							)				->   Matrix:






       """simple docstring"""
       ...




   @overload
   def __mul__(					self,	__magic_name__							)				->   Vector:






       """simple docstring"""
       ...




   def __mul__(					self,	__magic_name__							)				->   Vector | Matrix:






       """simple docstring"""
       if isinstance(__magic_name__,	__magic_name__							):  # matrix-vector
           if len(__magic_name__							) == self.__width:
               UpperCamelCase__							:       str      =   zero_vector(self.__height							)
               for i in range(self.__height							):
                   UpperCamelCase__							:       List[str]      =   [
                       self.__matrix[i][j] * other.component(__magic_name__							)
                       for j in range(self.__width							)
                   ]
                   ans.change_component(__magic_name__,	sum(__magic_name__							)							)
               return ans
           else:
               raise Exception(
                   '''vector must have the same size as the '''
                   '''number of columns of the matrix!'''							)
       elif isinstance(__magic_name__,	(int, float)							):  # matrix-scalar
           UpperCamelCase__							:       Tuple      =   [
               [self.__matrix[i][j] * other for j in range(self.__width							)]
               for i in range(self.__height							)
           ]
           return Matrix(__magic_name__,	self.__width,	self.__height							)
       return None




   def    UpperCamelCase__						(					self							)				->   int:






       """simple docstring"""
       return self.__height




   def    UpperCamelCase__						(					self							)				->   int:






       """simple docstring"""
       return self.__width




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__							)				->   float:






       """simple docstring"""
       if 0 <= x < self.__height and 0 <= y < self.__width:
           return self.__matrix[x][y]
       else:
           raise Exception('''change_component: indices out of bounds'''							)




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__,	__magic_name__							)				->   None:






       """simple docstring"""
       if 0 <= x < self.__height and 0 <= y < self.__width:
           UpperCamelCase__							:       str      =   value
       else:
           raise Exception('''change_component: indices out of bounds'''							)




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__							)				->   float:






       """simple docstring"""
       if self.__height != self.__width:
           raise Exception('''Matrix is not square'''							)
       UpperCamelCase__							:       int      =   self.__matrix[:x] + self.__matrix[x + 1 :]
       for i in range(len(__magic_name__							)							):
           UpperCamelCase__							:       Optional[Any]      =   minor[i][:y] + minor[i][y + 1 :]
       return Matrix(__magic_name__,	self.__width - 1,	self.__height - 1							).determinant()




   def    UpperCamelCase__						(					self,	__magic_name__,	__magic_name__							)				->   float:






       """simple docstring"""
       if self.__height != self.__width:
           raise Exception('''Matrix is not square'''							)
       if 0 <= x < self.__height and 0 <= y < self.__width:
           return (-1) ** (x + y) * self.minor(__magic_name__,	__magic_name__							)
       else:
           raise Exception('''Indices out of bounds'''							)




   def    UpperCamelCase__						(					self							)				->   float:






       """simple docstring"""
       if self.__height != self.__width:
           raise Exception('''Matrix is not square'''							)
       if self.__height < 1:
           raise Exception('''Matrix has no element'''							)
       elif self.__height == 1:
           return self.__matrix[0][0]
       elif self.__height == 2:
           return (
               self.__matrix[0][0] * self.__matrix[1][1]
               - self.__matrix[0][1] * self.__matrix[1][0]
           )
       else:
           UpperCamelCase__							:       Optional[Any]      =   [
               self.__matrix[0][y] * self.cofactor(0,	__magic_name__							) for y in range(self.__width							)
           ]
           return sum(__magic_name__							)




def 	lowerCAmelCase_					(						__UpperCAmelCase:    int   )   ->			Matrix:
    UpperCamelCase__							:       list[list[float]]      =   [[0] * n for _ in range(__UpperCAmelCase   )]
    return Matrix(__UpperCAmelCase	,     __UpperCAmelCase	,     __UpperCAmelCase   )





def 	lowerCAmelCase_					(						__UpperCAmelCase:    int	,     __UpperCAmelCase:    int	,     __UpperCAmelCase:    int	,     __UpperCAmelCase:    int   )   ->			Matrix:
    random.seed(__UpperCAmelCase   )
    UpperCamelCase__							:       list[list[float]]      =   [
        [random.randint(__UpperCAmelCase	,     __UpperCAmelCase   ) for _ in range(__UpperCAmelCase   )] for _ in range(__UpperCAmelCase   )
    ]
    return Matrix(__UpperCAmelCase	,     __UpperCAmelCase	,     __UpperCAmelCase   )








def 	lowerCAmelCase_					(						__UpperCAmelCase:    dict   )   ->			set:
    UpperCamelCase__							:       int      =   set()
    # edges = list of graph's edges
    UpperCamelCase__							:       str      =   get_edges(__UpperCAmelCase   )

    # While there are still elements in edges list, take an arbitrary edge
    # (from_node, to_node) and add his extremity to chosen_vertices and then
    # remove all arcs adjacent to the from_node and to_node
    while edges:
        UpperCamelCase__     ,UpperCamelCase__							:       List[Any]      =   edges.pop()
        chosen_vertices.add(__UpperCAmelCase   )
        chosen_vertices.add(__UpperCAmelCase   )
        for edge in edges.copy():
            if from_node in edge or to_node in edge:
                edges.discard(__UpperCAmelCase   )
    return chosen_vertices




def 	lowerCAmelCase_					(						__UpperCAmelCase:    dict   )   ->			set:
    UpperCamelCase__							:       Optional[int]      =   set()
    for from_node, to_nodes in graph.items():
        for to_node in to_nodes:
            edges.add((from_node, to_node)   )
    return edges


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  # graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
  # print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")

