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)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Text;



class A
{
    static int n;
    public static void Main() {
        n = int.Parse(Console.ReadLine());
        int[] nodes = new int[n+1];
        int[] data = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
        data.CopyTo(nodes, 1);
        for (int i = n / 2; i > 0; i--) {
            Maxheapify(nodes, i);
        }
        for(int i=1;i<=n;i++) {
            Console.Write(" " +nodes[i]);
        }
        Console.WriteLine();
    }
    public static void Maxheapify(int[] nodes, int i) {
        int left = i * 2;
        int right = i * 2 + 1;
        int largest = (left <= n && nodes[left] > nodes[i]) ? left : i;
        if (right <= n && nodes[right] > nodes[largest]) {
            largest = right;
        }
        if (largest != i) {
            int temp = nodes[i];
            nodes[i] = nodes[largest];
            nodes[largest] = temp;
            Maxheapify(nodes, largest);
        }
    }
}