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.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.StringTokenizer;

class Main{
    public static void main(String[] Args) throws IOException{
        final BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        final int n = Integer.parseInt(br.readLine());
        final int[] h = new int[n+1];
        final StringTokenizer st = new StringTokenizer(br.readLine(), " ");
        for(int i = 1; i<=n; i++){
            h[i] = Integer.parseInt(st.nextToken());
        }
        buildMaxHeap(h, n);
        final StringBuilder sb = new StringBuilder();
        for(int i = 1; i<=n; i++)
            sb.append(' ').append(h[i]);
        System.out.println(sb);
    }
    private static void buildMaxHeap(int[] h, int n){
        for(int i = n>>>1; i>0; i--)
            maxHeapify(h, i, n);
    }
    private static void maxHeapify(int[] a, int i, int n){
        int learge = i;
        do{
            i = learge;
            final int l = i+i;
            final int r = l+1;
            if(l<=n&&a[i]<a[l]) learge = l;
            if(r<=n&&a[learge]<a[r]) learge = r;
            if(i!=learge){
                final int t = a[i];
                a[i] = a[learge];
                a[learge] = t;
                maxHeapify(a, learge, n);
            }
        }while(i!=learge);
    }
}