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 JAVA (judge verdict: Accepted)

import java.io.*;
import java.util.*;

public class Main{
    public static void main(String args[]) throws IOException{
	BufferedReader br=new BufferedReader(new InputStreamReader(System.in));

	int h=Integer.parseInt(br.readLine());

	int a[]=new int[h+1];
	String str[]=br.readLine().split(" ");
	
	for(int i=0;i<h;i++){
	    a[i+1]=Integer.parseInt(str[i]);
	}

	for(int i=h/2;i>=1;i--){
	    maxHeapify(a,h,i);
	}

	for(int i=1;i<=h;i++){
	    System.out.print(" "+a[i]);
	}
	System.out.println();
    }

    static void maxHeapify(int a[],int h,int i){
	    int left=getLeft(i);
	    int right=getRight(i);

	    int max=0;

	    if(left<=h && a[left]>a[i]){
		max=left;
	    }else{
		max=i;
	    }

	    if(right<=h && a[right]>a[max]){
		max=right;
	    }

	    if(max!=i){
		int temp=a[i];
		a[i]=a[max];
		a[max]=temp;
		maxHeapify(a,h,max);	
	    }
    }
    
    static int getParent(int i){
	return i/2;
    }

    static int getLeft(int i){
	return 2*i;
    }

    static int getRight(int i){
	return 2*i+1;
    }
}