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)

/*
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 * To change this template file, choose Tools | Templates
 * and open the template in the editor.
 */

/* 
 * File:   main.c
 * Author: Ivan
 *
 * Created on July 19, 2018, 2:10 AM
 */

#include <stdio.h>
#include <stdlib.h>
#define MAX 5000001

/*
 * 
 */

int H, Heap[MAX];

void maxHeapify(int i);
void buildHeap();

int main(int argc, char** argv) {
    scanf("%d", &H);
    for (int i = 1; i <= H; ++i) scanf("%d", &Heap[i]);
    buildHeap();
    for (int i = 1; i <= H; ++i) printf(" %d", Heap[i]);
    printf("\n");
    return (EXIT_SUCCESS);
}

void maxHeapify(int i)
{
    int left, right, max, temp;
    left = i*2;
    right = i*2+1;
    
    if (left <= H && Heap[left] > Heap[i]) max = left;
    else max = i;
    if (right <= H && Heap[right] > Heap[max]) max = right;
    
    if (max != i)
    {
        temp = Heap[i];
        Heap[i] = Heap[max];
        Heap[max] = temp;
        maxHeapify(max);
    }
}

void buildHeap()
{
    for (int i = H/2; i >= 1; --i) maxHeapify(i);
}