"""simple docstring"""



class 				snake_case__     :


 def __init__(				self  ,						lowerCamelCase  ,						lowerCamelCase  ,						lowerCamelCase		):
        __a     				=  None
        __a     				=  None
        __a     				=  graph

        self._normalize_graph(__snake_case  ,						__snake_case		)
        __a     				=  len(__snake_case		)
        __a     				=  None


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase		):
        if sources is int:
               __a     				=  [sources]
        if sinks is int:
               __a     				=  [sinks]

        if len(__snake_case		) == 0 or len(__snake_case		) == 0:
               return

        __a     				=  sources[0]
        __a     				=  sinks[0]

        # make fake vertex if there are more
        # than one source or sink
        if len(__snake_case		) > 1 or len(__snake_case		) > 1:
               __a     				=  0
               for i in sources:
                      max_input_flow += sum(self.graph[i]		)

               __a     				=  len(self.graph		) + 1
               for room in self.graph:
                      room.insert(0  ,						0		)
               self.graph.insert(0  ,						[0] * size		)
               for i in sources:
                      __a     				=  max_input_flow
               __a     				=  0

               __a     				=  len(self.graph		) + 1
               for room in self.graph:
                      room.append(0		)
               self.graph.append([0] * size		)
               for i in sinks:
                      __a     				=  max_input_flow
               __a     				=  size - 1


 def 		a__ (				self		):
        if self.maximum_flow_algorithm is None:
               raise Exception("You need to set maximum flow algorithm before."		)
        if self.source_index is None or self.sink_index is None:
               return 0

        self.maximum_flow_algorithm.execute()
        return self.maximum_flow_algorithm.getMaximumFlow()






 def 		a__ (				self  ,						lowerCamelCase		):
        __a     				=  algorithm(self		)






class 				snake_case__     :


 def __init__(				self  ,						lowerCamelCase		):
        __a     				=  flow_network
        __a     				=  flow_network.verticesCount
        __a     				=  flow_network.sourceIndex
        __a     				=  flow_network.sinkIndex
        # it's just a reference, so you shouldn't change
        # it in your algorithms, use deep copy before doing that
        __a     				=  flow_network.graph
        __a     				=  False


 def 		a__ (				self		):
        if not self.executed:
               self._algorithm()
               __a     				=  True






 def 		a__ (				self		):
        pass






class 				snake_case__     (   A__   ):


 def __init__(				self  ,						lowerCamelCase		):
        super().__init__(__snake_case		)
        # use this to save your result
        __a     				=  -1






 def 		a__ (				self		):
        if not self.executed:
               raise Exception("You should execute algorithm before using its result!"		)

        return self.maximum_flow






class 				snake_case__     (   A__   ):


 def __init__(				self  ,						lowerCamelCase		):
        super().__init__(__snake_case		)

        __a     				=  [[0] * self.verticies_count for i in range(self.verticies_count		)]

        __a     				=  [0] * self.verticies_count
        __a     				=  [0] * self.verticies_count


 def 		a__ (				self		):
        __a     				=  self.verticies_count

        # push some substance to graph
        for nextvertex_index, bandwidth in enumerate(self.graph[self.source_index]		):
               self.preflow[self.source_index][nextvertex_index] += bandwidth
               self.preflow[nextvertex_index][self.source_index] -= bandwidth
               self.excesses[nextvertex_index] += bandwidth

        # Relabel-to-front selection rule
        __a     				=  [
            i
            for i in range(self.verticies_count		)
            if i != self.source_index and i != self.sink_index
        ]

        # move through list
        __a     				=  0
        while i < len(__snake_case		):
               __a     				=  vertices_list[i]
               __a     				=  self.heights[vertex_index]
               self.process_vertex(__snake_case		)
               if self.heights[vertex_index] > previous_height:
                      # if it was relabeled, swap elements
                      # and start from 0 index
                      vertices_list.insert(0  ,						vertices_list.pop(__snake_case		)		)
                      __a     				=  0
               else:
                      i += 1

        __a     				=  sum(self.preflow[self.source_index]		)


 def 		a__ (				self  ,						lowerCamelCase		):
        while self.excesses[vertex_index] > 0:
               for neighbour_index in range(self.verticies_count		):
                      # if it's neighbour and current vertex is higher
                      if (
                          self.graph[vertex_index][neighbour_index]
                          - self.preflow[vertex_index][neighbour_index]
                          > 0
                          and self.heights[vertex_index] > self.heights[neighbour_index]
                      ):
                             self.push(__snake_case  ,						__snake_case		)

               self.relabel(__snake_case		)


 def 		a__ (				self  ,						lowerCamelCase  ,						lowerCamelCase		):
        __a     				=  min(
            self.excesses[from_index]  ,						self.graph[from_index][to_index] - self.preflow[from_index][to_index]  ,						)
        self.preflow[from_index][to_index] += preflow_delta
        self.preflow[to_index][from_index] -= preflow_delta
        self.excesses[from_index] -= preflow_delta
        self.excesses[to_index] += preflow_delta






 def 		a__ (				self  ,						lowerCamelCase		):
        __a     				=  None
        for to_index in range(self.verticies_count		):
               if (
                   self.graph[vertex_index][to_index]
                   - self.preflow[vertex_index][to_index]
                   > 0
               ) and (min_height is None or self.heights[to_index] < min_height):
                      __a     				=  self.heights[to_index]

        if min_height is not None:
               __a     				=  min_height + 1


if __name__ == "__main__":
 SCREAMING_SNAKE_CASE__:Union[str, Any]			=	[0]
 SCREAMING_SNAKE_CASE__:str			=	[3]
 # graph = [
 #     [0, 0, 4, 6, 0, 0],
 #     [0, 0, 5, 2, 0, 0],
 #     [0, 0, 0, 0, 4, 4],
 #     [0, 0, 0, 0, 6, 6],
 #     [0, 0, 0, 0, 0, 0],
 #     [0, 0, 0, 0, 0, 0],
 # ]
 SCREAMING_SNAKE_CASE__:List[Any]			=	[[0, 7, 0, 0], [0, 0, 6, 0], [0, 0, 0, 8], [9, 0, 0, 0]]

 # prepare our network
 SCREAMING_SNAKE_CASE__:Tuple			=	FlowNetwork(graph, entrances, exits)
 # set algorithm
 flow_network.set_maximum_flow_algorithm(PushRelabelExecutor)
 # and calculate
 SCREAMING_SNAKE_CASE__:Any			=	flow_network.find_maximum_flow()

 print(F'''maximum flow is {maximum_flow}''')

"""simple docstring"""



from __future__ import annotations



def 					_lowerCamelCase(					a ,  a ,  a ,  a ,  a ,  ):
       __a     				=  len(a							)

       # 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(a							):
              # 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] ,  a ,  a ,  )



def 					_lowerCamelCase(					a							):
       __a     				=  []
       depth_first_search([] ,  [] ,  [] ,  a ,  a							)

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

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


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   n_queens_solution(4)
