
"""simple docstring"""

from __future__ import annotations

from typing import Any

class __lowerCAmelCase		:





     def __init__(     self     ,					__UpperCAmelCase     ,					__UpperCAmelCase     ,					__UpperCAmelCase = 0				):

      '''simple docstring'''




      __UpperCamelCase  ,     __UpperCamelCase      =	row, column
      __UpperCamelCase      =	[[default_value for c in range(__UpperCAmelCase				)] for r in range(__UpperCAmelCase				)]





     def __str__(     self				):

      '''simple docstring'''




      __UpperCamelCase      =	F'Matrix consist of {self.row} rows and {self.column} columns\n'

      # Make string identifier
      __UpperCamelCase      =	0
      for row_vector in self.array:
       for obj in row_vector:
        __UpperCamelCase      =	max(__UpperCAmelCase     ,					len(str(__UpperCAmelCase				)				)				)
      __UpperCamelCase      =	F'%{max_element_length}s'

      # Make string and return
      def single_line(__UpperCAmelCase				) -> str:
       nonlocal string_format_identifier
       __UpperCamelCase      =	'['
       line += ", ".join(string_format_identifier % (obj,) for obj in row_vector				)
       line += "]"
       return line

      s += "\n".join(single_line(__UpperCAmelCase				) for row_vector in self.array				)
      return s





     def __repr__(     self				):

      '''simple docstring'''




      return str(self				)





     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      if not (isinstance(__UpperCAmelCase     ,					(list, tuple)				) and len(__UpperCAmelCase				) == 2):
       return False
      elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
       return False
      else:
       return True





     def __getitem__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      assert self.validate_indicies(__UpperCAmelCase				)
      return self.array[loc[0]][loc[1]]





     def __setitem__(     self     ,					__UpperCAmelCase     ,					__UpperCAmelCase				):

      '''simple docstring'''




      assert self.validate_indicies(__UpperCAmelCase				)
      __UpperCamelCase      =	value





     def __add__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      assert isinstance(__UpperCAmelCase     ,					__UpperCAmelCase				)
      assert self.row == another.row and self.column == another.column

      # Add
      __UpperCamelCase      =	Matrix(self.row     ,					self.column				)
      for r in range(self.row				):
       for c in range(self.column				):
        __UpperCamelCase      =	self[r, c] + another[r, c]
      return result





     def __neg__(     self				):

      '''simple docstring'''




      __UpperCamelCase      =	Matrix(self.row     ,					self.column				)
      for r in range(self.row				):
       for c in range(self.column				):
        __UpperCamelCase      =	-self[r, c]
      return result





     def __sub__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      return self + (-another)





     def __mul__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      if isinstance(__UpperCAmelCase     ,					(int, float)				):  # Scalar multiplication
       __UpperCamelCase      =	Matrix(self.row     ,					self.column				)
       for r in range(self.row				):
        for c in range(self.column				):
         __UpperCamelCase      =	self[r, c] * another
       return result
      elif isinstance(__UpperCAmelCase     ,					__UpperCAmelCase				):  # Matrix multiplication
       assert self.column == another.row
       __UpperCamelCase      =	Matrix(self.row     ,					another.column				)
       for r in range(self.row				):
        for c in range(another.column				):
         for i in range(self.column				):
          result[r, c] += self[r, i] * another[i, c]
       return result
      else:
       __UpperCamelCase      =	F'Unsupported type given for another ({type(__UpperCAmelCase				)})'
       raise TypeError(__UpperCAmelCase				)





     def 						UpperCAmelCase (     self				):

      '''simple docstring'''




      __UpperCamelCase      =	Matrix(self.column     ,					self.row				)
      for r in range(self.row				):
       for c in range(self.column				):
        __UpperCamelCase      =	self[r, c]
      return result







     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase     ,					__UpperCAmelCase				):

      '''simple docstring'''




      assert isinstance(__UpperCAmelCase     ,					__UpperCAmelCase				) and isinstance(__UpperCAmelCase     ,					__UpperCAmelCase				)
      assert self.row == self.column == u.row == v.row  # u, v should be column vector
      assert u.column == v.column == 1  # u, v should be column vector

      # Calculate
      __UpperCamelCase      =	v.transpose()
      __UpperCamelCase      =	(v_t * self * u)[0, 0] + 1
      if numerator_factor == 0:
       return None  # It's not invertable
      return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":






      def 							A						(    )				->     None:
       # a^(-1)
       __UpperCamelCase      =	Matrix(3    , 3    , 0		)
       for i in range(3		):
        __UpperCamelCase      =	1
       print(f'a^(-1) is {ainv}'		)
       # u, v
       __UpperCamelCase      =	Matrix(3    , 1    , 0		)
       __UpperCamelCase  ,     __UpperCamelCase  ,     __UpperCamelCase      =	1, 2, -3
       __UpperCamelCase      =	Matrix(3    , 1    , 0		)
       __UpperCamelCase  ,     __UpperCamelCase  ,     __UpperCamelCase      =	4, -2, 5
       print(f'u is {u}'		)
       print(f'v is {v}'		)
       print(f'uv^T is {u * v.transpose()}'		)
       # Sherman Morrison
       print(f'(a + uv^T)^(-1) is {ainv.sherman_morrison(snake_case    , snake_case		)}'		)






      def 							A						(    )				->     None:
       import doctest

       doctest.testmod()





      testa()





"""simple docstring"""

import heapq as hq
import math
from collections.abc import Iterator

class __lowerCAmelCase		:





     def __init__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      __UpperCamelCase      =	str(id_				)
      __UpperCamelCase      =	None
      __UpperCamelCase      =	None
      __UpperCamelCase      =	[]
      __UpperCamelCase      =	{}  # {vertex:distance}





     def __lt__(     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      return self.key < other.key





     def __repr__(     self				):

      '''simple docstring'''




      return self.id





     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase				):

      '''simple docstring'''




      self.neighbors.append(__UpperCAmelCase				)







     def 						UpperCAmelCase (     self     ,					__UpperCAmelCase     ,					__UpperCAmelCase				):

      '''simple docstring'''




      __UpperCamelCase      =	weight






def 							A						(    snake_case				:List[Any]    , snake_case				:Dict    , snake_case				:Any    , snake_case				:str		)				->     List[str]:
 # add the neighbors:
 graph[a - 1].add_neighbor(graph[b - 1]		)
 graph[b - 1].add_neighbor(graph[a - 1]		)
 # add the edges:
 graph[a - 1].add_edge(graph[b - 1]    , snake_case		)
 graph[b - 1].add_edge(graph[a - 1]    , snake_case		)






def 							A						(    snake_case				:list    , snake_case				:Vertex		)				->     list:
 __UpperCamelCase      =	[]
 for u in graph:
  __UpperCamelCase      =	math.inf
  __UpperCamelCase      =	None
 __UpperCamelCase      =	0
 __UpperCamelCase      =	graph[:]
 while q:
  __UpperCamelCase      =	min(snake_case		)
  q.remove(snake_case		)
  for v in u.neighbors:
   if (v in q) and (u.edges[v.id] < v.key):
    __UpperCamelCase      =	u
    __UpperCamelCase      =	u.edges[v.id]
 for i in range(1    , len(snake_case		)		):
  a.append((int(graph[i].id		) + 1, int(graph[i].pi.id		) + 1)		)
 return a






def 							A						(    snake_case				:list    , snake_case				:Vertex		)				->     Iterator[tuple]:
 for u in graph:
  __UpperCamelCase      =	math.inf
  __UpperCamelCase      =	None
 __UpperCamelCase      =	0

 __UpperCamelCase      =	list(snake_case		)
 hq.heapify(snake_case		)

 while h:
  __UpperCamelCase      =	hq.heappop(snake_case		)
  for v in u.neighbors:
   if (v in h) and (u.edges[v.id] < v.key):
    __UpperCamelCase      =	u
    __UpperCamelCase      =	u.edges[v.id]
    hq.heapify(snake_case		)

 for i in range(1    , len(snake_case		)		):
  yield (int(graph[i].id		) + 1, int(graph[i].pi.id		) + 1)






def 							A						(    )				->     None:
 pass


if __name__ == "__main__":
      import doctest

      doctest.testmod()



