Problem:
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Maximum Heap

A binary heap which satisfies max-heap property is called max-heap. In a max-heap, for every node $i$ other than the root, $A[i] \leq A[parent(i)]$, that is, the value of a node is at most the value of its parent. The largest element in a max-heap is stored at the root, and the subtree rooted at a node contains values no larger than that contained at the node itself.

Here is an example of a max-heap.

Write a program which reads an array and constructs a max-heap from the array based on the following pseudo code.

$maxHeapify(A, i)$ move the value of $A[i]$ down to leaves to make a sub-tree of node $i$ a max-heap. Here, $H$ is the size of the heap.

1 maxHeapify(A, i)
2 l = left(i)
3 r = right(i)
4 // select the node which has the maximum value
5 if l ≤ H and A[l] > A[i]
6 largest = l
7 else
8 largest = i
9 if r ≤ H and A[r] > A[largest]
10 largest = r
11
12 if largest ≠ i　// value of children is larger than that of i
13 swap A[i] and A[largest]
14 maxHeapify(A, largest) // call recursively

The following procedure buildMaxHeap(A) makes $A$ a max-heap by performing maxHeapify in a bottom-up manner.

1 buildMaxHeap(A)
2 for i = H/2 downto 1
3 maxHeapify(A, i)

Input

In the first line, an integer $H$ is given. In the second line, $H$ integers which represent elements in the binary heap are given in order of node id (from $1$ to $H$).

Output

Print values of nodes in the max-heap in order of their id (from $1$ to $H$). Print a single space character before each value.

Constraint

$1 \leq H \leq 500,000$

$-2,000,000,000 \leq$ value of a node $\leq 2,000,000,000$

Sample Input 1

10
4 1 3 2 16 9 10 14 8 7

Sample Output 1

16 14 10 8 7 9 3 2 4 1

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in C (judge verdict: Accepted)

/*
* Complete Binaly Tree
* output 1)node_number, 2)parent_number, 3)depth, 4)kind_of_node, 5)child_list
*/

/* inclution */
#include <stdio.h>

/* definition */
#define HEAP_SIZE_MAX 500000
#define VALUE_MAX 2000000000
#define VALUE_MIN -2000000000

/* Prototype */
int parent(int);
int left(int);
int right(int);
void maxHeapify(int);
void swap(int, int);


/* Global Variables */
int n, A[HEAP_SIZE_MAX+1];

/* ??????????????´?????§????§?????????? */
void maxHeapify(int i) {
  int l, r, largest;
  l = left(i);
  r = right(i);
  // ????????????????????????????????§???????????§????????????????????¶
  if(l<=n && A[l]>A[i]) { //?????????????????°???
    largest = l;
  }else {
    largest = i;
  }
  if(r<=n && A[r]>A[largest]) { //?¬????????????°??????????????¨?????????
    largest = r;
  }
  if(largest != i){ // i ???????????????????????§????????´???
    swap(i, largest);
    maxHeapify(largest); // ?????°??????????????????
  }
}

void swap(int i, int largest) {
  int tmp;  //A[i] ??¨ A[largest] ?????????;
    tmp = A[i];
    A[i] = A[largest];
    A[largest] = tmp;
}

/* node operation */
int parent(int i){
  return i/2;
}
int left(int i){
  return 2 * i;
}
int right(int i){
  return 2 * i + 1;
}

/* main */
int main(){
  int i;
  
  /* input the number of A[] */
  scanf("%d",&n);
  
  /* input array */
  for(i=1;i<=n;i++) {
    scanf("%d", &A[i]);
  }

  for(i=n/2;i>=1;i--){ // O(logn)
    maxHeapify(i);
  }

  /* print */
  for(i=1;i<=n;i++) {
    printf(" %d", A[i]);
  }
  printf("\n");

  return 0;
}