




from __future__ import annotations

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     int							) ->			list[list[int]]:
      SCREAMING_SNAKE_CASE_     						=      []
      create_all_state(1      ,      __UpperCAmelCase      ,      __UpperCAmelCase      ,      []      ,      __UpperCAmelCase							)
      return result

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     int      ,      __UpperCAmelCase						:     list[int]      ,      __UpperCAmelCase						:     list[list[int]]      ,      ) ->			None:
      if level == 0:
            total_list.append(current_list[:]							)
            return

      for i in range(__UpperCAmelCase      ,      total_number - level + 2							):
            current_list.append(__UpperCAmelCase							)
            create_all_state(i + 1      ,      __UpperCAmelCase      ,      level - 1      ,      __UpperCAmelCase      ,      __UpperCAmelCase							)
            current_list.pop()

def  UpperCAmelCase_							(				__UpperCAmelCase						:     list[list[int]]							) ->			None:
      for i in total_list:
            print(*__UpperCAmelCase							)


if __name__ == "__main__":
   lowerCamelCase__			:       Any         =							4
   lowerCamelCase__			:       Optional[Any]         =							2
   lowerCamelCase__			:       Optional[int]         =							generate_all_combinations(n, k)
   print_all_state(total_list)





from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

lowerCamelCase__			:       Dict         =							TypeVar('T')

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (position - 1) // 2

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (2 * position) + 1

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (2 * position) + 2





class 	lowerCamelCase_							(				Generic[T] ):


 '''simple docstring'''







 def __init__(							self  :		Optional[Any]						):
       SCREAMING_SNAKE_CASE_     						=      []
       SCREAMING_SNAKE_CASE_     						=      {}
       SCREAMING_SNAKE_CASE_     						=      0



 def __len__(							self  :		Optional[Any]						):
       return self.elements



 def __repr__(							self  :		Optional[int]						):
       return str(self.heap						)



 def 				lowerCAmelCase_  (							self  :		Union[str, Any]						):
       # Check if the priority queue is empty
       return self.elements == 0



 def 				lowerCAmelCase_  (							self  :		Tuple    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Add an element with given priority to the queue
       self.heap.append((elem, weight)						)
       SCREAMING_SNAKE_CASE_     						=      self.elements
       self.elements += 1
       self._bubble_up(_lowerCAmelCase						)



 def 				lowerCAmelCase_  (							self  :		Optional[int]						):
       # Remove and return the element with lowest weight (highest priority)
       if self.elements > 1:
             self._swap_nodes(0    ,    self.elements - 1						)
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap.pop()
       del self.position_map[elem]
       self.elements -= 1
       if self.elements > 0:
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[0]
             self._bubble_down(_lowerCAmelCase						)
       return elem



 def 				lowerCAmelCase_  (							self  :		Optional[int]    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Update the weight of the given key
       SCREAMING_SNAKE_CASE_     						=      self.position_map[elem]
       SCREAMING_SNAKE_CASE_     						=      (elem, weight)
       if position > 0:
             SCREAMING_SNAKE_CASE_     						=      get_parent_position(_lowerCAmelCase						)
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[parent_position]
             if parent_weight > weight:
                   self._bubble_up(_lowerCAmelCase						)
             else:
                   self._bubble_down(_lowerCAmelCase						)
       else:
             self._bubble_down(_lowerCAmelCase						)



 def 				lowerCAmelCase_  (							self  :		Union[str, Any]    ,    _lowerCAmelCase  :		T						):
       # Place a node at the proper position (upward movement) [to be used internally
       # only]
       SCREAMING_SNAKE_CASE_     						=      self.position_map[elem]
       if curr_pos == 0:
             return None
       SCREAMING_SNAKE_CASE_     						=      get_parent_position(_lowerCAmelCase						)
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[curr_pos]
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[parent_position]
       if parent_weight > weight:
             self._swap_nodes(_lowerCAmelCase    ,    _lowerCAmelCase						)
             return self._bubble_up(_lowerCAmelCase						)
       return None



 def 				lowerCAmelCase_  (							self  :		List[Any]    ,    _lowerCAmelCase  :		T						):
       # Place a node at the proper position (downward movement) [to be used
       # internally only]
       SCREAMING_SNAKE_CASE_     						=      self.position_map[elem]
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[curr_pos]
       SCREAMING_SNAKE_CASE_     						=      get_child_left_position(_lowerCAmelCase						)
       SCREAMING_SNAKE_CASE_     						=      get_child_right_position(_lowerCAmelCase						)
       if child_left_position < self.elements and child_right_position < self.elements:
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[child_left_position]
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[child_right_position]
             if child_right_weight < child_left_weight and child_right_weight < weight:
                   self._swap_nodes(_lowerCAmelCase    ,    _lowerCAmelCase						)
                   return self._bubble_down(_lowerCAmelCase						)
       if child_left_position < self.elements:
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[child_left_position]
             if child_left_weight < weight:
                   self._swap_nodes(_lowerCAmelCase    ,    _lowerCAmelCase						)
                   return self._bubble_down(_lowerCAmelCase						)
       else:
             return None
       if child_right_position < self.elements:
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[child_right_position]
             if child_right_weight < weight:
                   self._swap_nodes(_lowerCAmelCase    ,    _lowerCAmelCase						)
                   return self._bubble_down(_lowerCAmelCase						)
       return None




 def 				lowerCAmelCase_  (							self  :		Optional[Any]    ,    _lowerCAmelCase  :		int    ,    _lowerCAmelCase  :		int						):
       # Swap the nodes at the given positions
       SCREAMING_SNAKE_CASE_     						=      self.heap[nodea_pos][0]
       SCREAMING_SNAKE_CASE_     						=      self.heap[nodea_pos][0]
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      (
           self.heap[nodea_pos],
           self.heap[nodea_pos],
       )
       SCREAMING_SNAKE_CASE_     						=      nodea_pos
       SCREAMING_SNAKE_CASE_     						=      nodea_pos





class 	lowerCamelCase_							(				Generic[T] ):


 '''simple docstring'''







 def __init__(							self  :		Union[str, Any]						):
       SCREAMING_SNAKE_CASE_     						=      {}
       SCREAMING_SNAKE_CASE_     						=      0



 def __repr__(							self  :		Optional[int]						):
       return str(self.connections						)



 def __len__(							self  :		Tuple						):
       return self.nodes



 def 				lowerCAmelCase_  (							self  :		Dict    ,    _lowerCAmelCase  :		T						):
       # Add a node in the graph if it is not in the graph
       if node not in self.connections:
             SCREAMING_SNAKE_CASE_     						=      {}
             self.nodes += 1




 def 				lowerCAmelCase_  (							self  :		List[Any]    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Add an edge between 2 nodes in the graph
       self.add_node(_lowerCAmelCase						)
       self.add_node(_lowerCAmelCase						)
       SCREAMING_SNAKE_CASE_     						=      weight
       SCREAMING_SNAKE_CASE_     						=      weight


def  UpperCAmelCase_							(				__UpperCAmelCase						:     GraphUndirectedWeighted[T]      ,      ) ->			tuple[dict[T, int], dict[T, T | None]]:
      SCREAMING_SNAKE_CASE_     						=      {node: maxsize for node in graph.connections}
      SCREAMING_SNAKE_CASE_     						=      {node: None for node in graph.connections}

      SCREAMING_SNAKE_CASE_     						=      MinPriorityQueue()
      for node, weight in dist.items():
            priority_queue.push(__UpperCAmelCase      ,      __UpperCAmelCase							)

      if priority_queue.is_empty():
            return dist, parent

      # initialization
      SCREAMING_SNAKE_CASE_     						=      priority_queue.extract_min()
      SCREAMING_SNAKE_CASE_     						=      0
      for neighbour in graph.connections[node]:
            if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
                  SCREAMING_SNAKE_CASE_     						=      dist[node] + graph.connections[node][neighbour]
                  priority_queue.update_key(__UpperCAmelCase      ,      dist[neighbour]							)
                  SCREAMING_SNAKE_CASE_     						=      node

    # running prim's algorithm
      while not priority_queue.is_empty():
            SCREAMING_SNAKE_CASE_     						=      priority_queue.extract_min()
            for neighbour in graph.connections[node]:
                  if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
                        SCREAMING_SNAKE_CASE_     						=      dist[node] + graph.connections[node][neighbour]
                        priority_queue.update_key(__UpperCAmelCase      ,      dist[neighbour]							)
                        SCREAMING_SNAKE_CASE_     						=      node
      return dist, parent