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

public class Main {
	static int n, H[];
	
	public static void main(String[] args) {
		Scanner in = new Scanner(System.in);
		n=Integer.parseInt(in.next()); H=new int[n+1];
		for(int i=1;i<=n;i++) {
			H[i]=Integer.parseInt(in.next());
		}
		buildMaxHeap();
		
		PrintWriter out=new PrintWriter(System.out);
		for(int i=1;i<=n;i++)out.print(" "+H[i]);
		out.println();
		out.flush();
	}
	
	static void maxheapify(int i) {
		if(i*2>n) return;
		
		int max=H[i], u=-1;
		if(H[i*2]>max) {
			max=H[i*2];
			u=i*2;
		}
		if(i*2+1<=n && H[i*2+1]>max) {
			max=H[i*2+1];
			u=i*2+1;
		}
		
		if(u>0) {
			int t=H[i];
			H[i]=H[u];
			H[u]=t;
			maxheapify(u);
		}
	}
	
	static void buildMaxHeap() {
		for(int i=n/2;i>0;i--)maxheapify(i);
	}
}