






'''simple docstring'''
def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							):


   """simple docstring"""




   def update_area_of_max_square(_UpperCamelCase						,		_UpperCamelCase							) -> int:
      # BASE CASE
      if row >= rows or col >= cols:
         return 0

      lowercase_	:    List[str]           =     update_area_of_max_square(_UpperCamelCase						,		col + 1							)
      lowercase_	:    List[Any]           =     update_area_of_max_square(row + 1						,		col + 1							)
      lowercase_	:    Tuple           =     update_area_of_max_square(row + 1						,		_UpperCamelCase							)

      if mat[row][col]:
         lowercase_	:    Optional[int]           =     1 + min([right, diagonal, down]							)
         lowercase_	:    Any           =     max(largest_square_area[0]						,		_UpperCamelCase							)
         return sub_problem_sol
      else:
         return 0

   lowercase_	:    int           =     [0]
   update_area_of_max_square(0						,		0							)
   return largest_square_area[0]




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							):


   """simple docstring"""




   def update_area_of_max_square_using_dp_array(
       _UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							) -> int:
      if row >= rows or col >= cols:
         return 0
      if dp_array[row][col] != -1:
         return dp_array[row][col]

      lowercase_	:    Dict           =     update_area_of_max_square_using_dp_array(_UpperCamelCase						,		col + 1						,		_UpperCamelCase							)
      lowercase_	:    str           =     update_area_of_max_square_using_dp_array(row + 1						,		col + 1						,		_UpperCamelCase							)
      lowercase_	:    Optional[Any]           =     update_area_of_max_square_using_dp_array(row + 1						,		_UpperCamelCase						,		_UpperCamelCase							)

      if mat[row][col]:
         lowercase_	:    Tuple           =     1 + min([right, diagonal, down]							)
         lowercase_	:    int           =     max(largest_square_area[0]						,		_UpperCamelCase							)
         lowercase_	:    Dict           =     sub_problem_sol
         return sub_problem_sol
      else:
         return 0

   lowercase_	:    Any           =     [0]
   lowercase_	:    Optional[int]           =     [[-1] * cols for _ in range(_UpperCamelCase							)]
   update_area_of_max_square_using_dp_array(0						,		0						,		_UpperCamelCase							)

   return largest_square_area[0]




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							):


   """simple docstring"""




   lowercase_	:    str           =     [[0] * (cols + 1) for _ in range(rows + 1							)]
   lowercase_	:    List[Any]           =     0
   for row in range(rows - 1						,		-1						,		-1							):
      for col in range(cols - 1						,		-1						,		-1							):
         lowercase_	:    Tuple           =     dp_array[row][col + 1]
         lowercase_	:    List[str]           =     dp_array[row + 1][col + 1]
         lowercase_	:    List[Any]           =     dp_array[row + 1][col]

         if mat[row][col] == 1:
            lowercase_	:    Any           =     1 + min(_UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							)
            lowercase_	:    Any           =     max(dp_array[row][col]						,		_UpperCamelCase							)
         else:
            lowercase_	:    int           =     0

   return largest_square_area




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							):


   """simple docstring"""




   lowercase_	:    Optional[int]           =     [0] * (cols + 1)
   lowercase_	:    Union[str, Any]           =     [0] * (cols + 1)
   lowercase_	:    int           =     0
   for row in range(rows - 1						,		-1						,		-1							):
      for col in range(cols - 1						,		-1						,		-1							):
         lowercase_	:    Dict           =     current_row[col + 1]
         lowercase_	:    List[Any]           =     next_row[col + 1]
         lowercase_	:    Tuple           =     next_row[col]

         if mat[row][col] == 1:
            lowercase_	:    Dict           =     1 + min(_UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							)
            lowercase_	:    int           =     max(current_row[col]						,		_UpperCamelCase							)
         else:
            lowercase_	:    Tuple           =     0
      lowercase_	:    Optional[Any]           =     current_row

   return largest_square_area


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 print(largest_square_area_in_matrix_bottom_up(2, 2, [[1, 1], [1, 1]]))












'''simple docstring'''
import heapq




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase							):


   """simple docstring"""




   lowercase_	:    list[list]           =     []

   # for each node and his adjacency list add them and the rank of the node to queue
   # using heapq module the queue will be filled like a Priority Queue
   # heapq works with a min priority queue, so I used -1*len(v) to build it
   for key, value in graph.items():
      # O(log(n))
      heapq.heappush(_UpperCamelCase						,		[-1 * len(_UpperCamelCase							), (key, value)]							)

   # chosen_vertices = set of chosen vertices
   lowercase_	:    Optional[Any]           =     set()

   # while queue isn't empty and there are still edges
   #   (queue[0][0] is the rank of the node with max rank)
   while queue and queue[0][0] != 0:
      # extract vertex with max rank from queue and add it to chosen_vertices
      lowercase_	:    Any           =     heapq.heappop(_UpperCamelCase							)[1][0]
      chosen_vertices.add(_UpperCamelCase							)

      # Remove all arcs adjacent to argmax
      for elem in queue:
         # if v haven't adjacent node, skip
         if elem[0] == 0:
            continue
         # if argmax is reachable from elem
         # remove argmax from elem's adjacent list and update his rank
         if argmax in elem[1][1]:
            lowercase_	:    str           =     elem[1][1].index(_UpperCamelCase							)
            del elem[1][1][index]
            elem[0] += 1
        # re-order the queue
      heapq.heapify(_UpperCamelCase							)
   return chosen_vertices


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 UpperCamelCase__ =			{0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
 print(f"""Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}""")




