


import math
import sys


class 	a__						:

       def __init__( self   :  int     ):
            a_     :  Dict				      =     0
            a_     :  Optional[int]				      =     []
            a_     :  List[Any]				      =     {}  # To store the pos of node in array

       def     UpperCAmelCase( self   :  Union[str, Any]     ):
            return self.cur_size == 0

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  str     ):
            a_     :  str				      =     self.left(lowerCamelCase_     )
            a_     :  Optional[Any]				      =     self.right(lowerCamelCase_     )
            if lc < self.cur_size and self.array(lowerCamelCase_     )[0] < self.array(lowerCamelCase_     )[0]:
                 a_     :  str				      =     lc
            else:
                 a_     :  Union[str, Any]				      =     idx
            if rc < self.cur_size and self.array(lowerCamelCase_     )[0] < self.array(lowerCamelCase_     )[0]:
                 a_     :  int				      =     rc
            if smallest != idx:
                 self.swap(lowerCamelCase_   ,				lowerCamelCase_     )
                 self.min_heapify(lowerCamelCase_     )

       def     UpperCAmelCase( self   :  Optional[Any]   ,				lowerCamelCase_   :  Union[str, Any]     ):
            # Inserts a node into the Priority Queue
            a_     :  Optional[int]				      =     self.cur_size
            self.cur_size += 1
            self.array.append((sys.maxsize, tup[1])     )
            self.decrease_key((sys.maxsize, tup[1])   ,				tup[0]     )

       def     UpperCAmelCase( self   :  Union[str, Any]     ):
            # Removes and returns the min element at top of priority queue
            a_     :  List[Any]				      =     self.array[0][1]
            a_     :  str				      =     self.array[self.cur_size - 1]
            self.cur_size -= 1
            self.min_heapify(1     )
            del self.pos[min_node]
            return min_node

       def     UpperCAmelCase( self   :  List[str]   ,				lowerCamelCase_   :  int     ):
            # returns the index of left child
            return 2 * i + 1

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  Any     ):
            # returns the index of right child
            return 2 * i + 2

       def     UpperCAmelCase( self   :  Tuple   ,				lowerCamelCase_   :  str     ):
            # returns the index of parent
            return math.floor(i / 2     )

       def     UpperCAmelCase( self   :  Dict   ,				lowerCamelCase_   :  int   ,				lowerCamelCase_   :  Optional[int]     ):
            # swaps array elements at indices i and j
            # update the pos{}
            a_     :  Optional[Any]				      =     j
            a_     :  Optional[int]				      =     i
            a_     :  Optional[int]				      =     self.array[i]
            a_     :  Dict				      =     self.array[j]
            a_     :  List[Any]				      =     temp

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  Any   ,				lowerCamelCase_   :  Optional[Any]     ):
            a_     :  Tuple				      =     self.pos[tup[1]]
            # assuming the new_d is atmost old_d
            a_     :  List[str]				      =     (new_d, tup[1])
            while idx > 0 and self.array[self.par(lowerCamelCase_     )][0] > self.array[idx][0]:
                 self.swap(lowerCamelCase_   ,				self.par(lowerCamelCase_     )     )
                 a_     :  int				      =     self.par(lowerCamelCase_     )


class 	a__						:

       def __init__( self   :  List[Any]   ,				lowerCamelCase_   :  Union[str, Any]     ):
            a_     :  Dict				      =     {}  # To store graph: u -> (v,w)
            a_     :  List[str]				      =     num  # Number of nodes in graph
            # To store the distance from source vertex
            a_     :  Any				      =     [0] * self.num_nodes
            a_     :  Any				      =     [-1] * self.num_nodes  # To store the path

       def     UpperCAmelCase( self   :  str   ,				lowerCamelCase_   :  Optional[Any]   ,				lowerCamelCase_   :  Union[str, Any]   ,				lowerCamelCase_   :  str     ):
            #  Edge going from node u to v and v to u with weight w
            # u (w)-> v, v (w) -> u
            # Check if u already in graph
            if u in self.adjList:
                 self.adjList[u].append((v, w)     )
            else:
                 a_     :  Any				      =     [(v, w)]

            # Assuming undirected graph
            if v in self.adjList:
                 self.adjList[v].append((u, w)     )
            else:
                 a_     :  List[Any]				      =     [(u, w)]

       def     UpperCAmelCase( self   :  Optional[Any]     ):
            # u -> v(w)
            for u in self.adjList:
                 print(lowerCamelCase_   ,				"""->"""   ,				""" -> """.join(str(F'''{v}({w})'''     ) for v, w in self.adjList[u]     )     )

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  List[Any]     ):
            # Flush old junk values in par[]
            a_     :  int				      =     [-1] * self.num_nodes
            # src is the source node
            a_     :  List[str]				      =     0
            a_     :  int				      =     PriorityQueue()
            q.insert((0, src)     )  # (dist from src, node)
            for u in self.adjList:
                 if u != src:
                      a_     :  List[Any]				      =     sys.maxsize  # Infinity
                      a_     :  Optional[int]				      =     -1

            while not q.is_empty():
                 a_     :  Union[str, Any]				      =     q.extract_min()  # Returns node with the min dist from source
                 # Update the distance of all the neighbours of u and
                 # if their prev dist was INFINITY then push them in Q
                 for v, w in self.adjList[u]:
                      a_     :  Dict				      =     self.dist[u] + w
                      if self.dist[v] > new_dist:
                           if self.dist[v] == sys.maxsize:
                                q.insert((new_dist, v)     )
                           else:
                                q.decrease_key((self.dist[v], v)   ,				lowerCamelCase_     )
                           a_     :  Dict				      =     new_dist
                           a_     :  Dict				      =     u

        # Show the shortest distances from src
            self.show_distances(lowerCamelCase_     )

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  Any     ):
            print(F'''Distance from node: {src}'''     )
            for u in range(self.num_nodes     ):
                 print(F'''Node {u} has distance: {self.dist[u]}'''     )

       def     UpperCAmelCase( self   :  Optional[Any]   ,				lowerCamelCase_   :  Dict   ,				lowerCamelCase_   :  int     ):
            # To show the shortest path from src to dest
            # WARNING: Use it *after* calling dijkstra
            a_     :  int				      =     []
            a_     :  Optional[int]				      =     0
            a_     :  Any				      =     dest
            # Backtracking from dest to src
            while self.par[temp] != -1:
                 path.append(lowerCamelCase_     )
                 if temp != src:
                      for v, w in self.adjList[temp]:
                           if v == self.par[temp]:
                                cost += w
                                break
                 a_     :  Dict				      =     self.par[temp]
            path.append(lowerCamelCase_     )
            path.reverse()

            print(F'''----Path to reach {dest} from {src}----'''     )
            for u in path:
                 print(F'''{u}'''   ,				end=""" """     )
                 if u != dest:
                      print("""-> """   ,				end=""""""     )

            print("""\nTotal cost of path: """   ,				lowerCamelCase_     )


if __name__ == "__main__":
       __lowerCamelCase   	=  Graph(9)
       graph.add_edge(0, 1, 4)
       graph.add_edge(0, 7, 8)
       graph.add_edge(1, 2, 8)
       graph.add_edge(1, 7, 11)
       graph.add_edge(2, 3, 7)
       graph.add_edge(2, 8, 2)
       graph.add_edge(2, 5, 4)
       graph.add_edge(3, 4, 9)
       graph.add_edge(3, 5, 14)
       graph.add_edge(4, 5, 10)
       graph.add_edge(5, 6, 2)
       graph.add_edge(6, 7, 1)
       graph.add_edge(6, 8, 6)
       graph.add_edge(7, 8, 7)
       graph.show_graph()
       graph.dijkstra(0)
       graph.show_path(0, 4)

# OUTPUT
# 0 -> 1(4) -> 7(8)
# 1 -> 0(4) -> 2(8) -> 7(11)
# 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7)
# 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4)
# 3 -> 2(7) -> 4(9) -> 5(14)
# 8 -> 2(2) -> 6(6) -> 7(7)
# 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2)
# 4 -> 3(9) -> 5(10)
# 6 -> 5(2) -> 7(1) -> 8(6)
# Distance from node: 0
# Node 0 has distance: 0
# Node 1 has distance: 4
# Node 2 has distance: 12
# Node 3 has distance: 19
# Node 4 has distance: 21
# Node 5 has distance: 11
# Node 6 has distance: 9
# Node 7 has distance: 8
# Node 8 has distance: 14
# ----Path to reach 4 from 0----
# 0 -> 7 -> 6 -> 5 -> 4
# Total cost of path:  21







from __future__ import annotations

from abc import abstractmethod
from collections.abc import Iterable
from typing import Generic, Protocol, TypeVar


class 	a__						(    lowerCAmelCase_     ):

       @abstractmethod
       def __lt__( self   :  T   ,				lowerCamelCase_   :  T     ):
            pass

       @abstractmethod
       def __gt__( self   :  T   ,				lowerCamelCase_   :  T     ):
            pass

       @abstractmethod
       def __eq__( self   :  T   ,				lowerCamelCase_   :  object     ):
            pass

__lowerCamelCase   	=  TypeVar('''T''', bound=Comparable)


class 	a__						(    Generic[T]     ):

       def __init__( self   :  Tuple     ):
            a_     :  list[T]				      =     []
            a_     :  int				      =     0

       def __repr__( self   :  Optional[Any]     ):
            return str(self.h     )

       def     UpperCAmelCase( self   :  Dict   ,				lowerCamelCase_   :  int     ):
            if child_idx > 0:
                 return (child_idx - 1) // 2
            return None

       def     UpperCAmelCase( self   :  Any   ,				lowerCamelCase_   :  int     ):
            a_     :  Tuple				      =     2 * parent_idx + 1
            if left_child_index < self.heap_size:
                 return left_child_index
            return None

       def     UpperCAmelCase( self   :  List[Any]   ,				lowerCamelCase_   :  int     ):
            a_     :  List[Any]				      =     2 * parent_idx + 2
            if right_child_index < self.heap_size:
                 return right_child_index
            return None

       def     UpperCAmelCase( self   :  Optional[int]   ,				lowerCamelCase_   :  int     ):
            if index < self.heap_size:
                 a_     :  int				      =     index
                 a_     :  Union[str, Any]				      =     self.left_child_idx(lowerCamelCase_     )
                 a_     :  Optional[int]				      =     self.right_child_idx(lowerCamelCase_     )
                 # check which child is larger than its parent
                 if left_child is not None and self.h[left_child] > self.h[violation]:
                      a_     :  str				      =     left_child
                 if right_child is not None and self.h[right_child] > self.h[violation]:
                      a_     :  List[Any]				      =     right_child
                 # if violation indeed exists
                 if violation != index:
                      # swap to fix the violation
                      a_      ,      a_     :  int				      =     self.h[index], self.h[violation]
                      # fix the subsequent violation recursively if any
                      self.max_heapify(lowerCamelCase_     )

       def     UpperCAmelCase( self   :  List[str]   ,				lowerCamelCase_   :  Iterable[T]     ):
            a_     :  Dict				      =     list(lowerCamelCase_     )
            a_     :  int				      =     len(self.h     )
            if self.heap_size > 1:
                 # max_heapify from right to left but exclude leaves (last level)
                 for i in range(self.heap_size // 2 - 1   ,				-1   ,				-1     ):
                      self.max_heapify(lowerCamelCase_     )

       def     UpperCAmelCase( self   :  Tuple     ):
            if self.heap_size >= 2:
                 a_     :  int				      =     self.h[0]
                 a_     :  Dict				      =     self.h.pop(-1     )
                 self.heap_size -= 1
                 self.max_heapify(0     )
                 return me
            elif self.heap_size == 1:
                 self.heap_size -= 1
                 return self.h.pop(-1     )
            else:
                 raise Exception("""Empty heap"""     )

       def     UpperCAmelCase( self   :  List[str]   ,				lowerCamelCase_   :  T     ):
            self.h.append(lowerCamelCase_     )
            a_     :  Optional[int]				      =     (self.heap_size - 1) // 2
            self.heap_size += 1
            while idx >= 0:
                 self.max_heapify(lowerCamelCase_     )
                 a_     :  Union[str, Any]				      =     (idx - 1) // 2

       def     UpperCAmelCase( self   :  int     ):
            a_     :  Union[str, Any]				      =     self.heap_size
            for j in range(size - 1   ,				0   ,				-1     ):
                 a_      ,      a_     :  str				      =     self.h[j], self.h[0]
                 self.heap_size -= 1
                 self.max_heapify(0     )
            a_     :  Union[str, Any]				      =     size


if __name__ == "__main__":
       import doctest

       # run doc test
       doctest.testmod()

       # demo
       for unsorted in [
           [0],
           [2],
           [3, 5],
           [5, 3],
           [5, 5],
           [0, 0, 0, 0],
           [1, 1, 1, 1],
           [2, 2, 3, 5],
           [0, 2, 2, 3, 5],
           [2, 5, 3, 0, 2, 3, 0, 3],
           [6, 1, 2, 7, 9, 3, 4, 5, 10, 8],
           [103, 9, 1, 7, 11, 15, 25, 201, 209, 107, 5],
           [-45, -2, -5],
       ]:
              print(f'''unsorted array: {unsorted}''')

              __lowerCamelCase   	=  Heap()
              heap.build_max_heap(unsorted)
              print(f'''after build heap: {heap}''')

              print(f'''max value: {heap.extract_max()}''')
              print(f'''after max value removed: {heap}''')

              heap.insert(100)
              print(f'''after new value 100 inserted: {heap}''')

              heap.heap_sort()
              print(f'''heap-sorted array: {heap}\n''')



