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.IO;
using System.Text;
using System.Collections;
using System.Collections.Generic;
 
class Program
{
    static void Main()
    {
        Console.ReadLine();
        int[] inputs = Console.ReadLine().Split(' ')
                                    .Select(val => int.Parse(val))
                                    .ToArray();
        Heap maxHeap = new Heap(inputs);
        maxHeap.BuildMaxHeap();
        maxHeap.Display();
    }
}

class Heap
{
    private const int MaxSize = 500000;
    public int Count {get; set;} // = 0;
    private int[] elements;
    public int this[int i]
    {
        get {return elements[i];}
        set {elements[i] = value;}
    }
    public Heap() : this(MaxSize) {}
    public Heap(int size)
    {
        elements = new int[size];
        Count = size;
    }
    public Heap(int[] array)
    {
        Count = array.Length;
        elements = new int[Count];
        for (int i = 0; i < Count; i++)
        {
            elements[i] = array[i];
        }
    }
    private int LeftOf(int i)
    {
        int val = 2 * (i + 1) - 1;
        return val < Count ? val : -1;
    }
    private int RightOf(int i)
    {
        int val = 2 * (i + 1);
        return val < Count ? val : -1;
    }
    public void BuildMaxHeap()
    {
        for (int i = Count / 2 - 1; i >= 0 ; i--)
        {
            Modify(i);
        }
    }
    private void Modify(int i)
    {
        int l = LeftOf(i);
        int r = RightOf(i);
        int largest = i;
        if(l != -1 && elements[l] > elements[i]) largest = l;
        if(r != -1 && elements[r] > elements[largest]) largest = r;
        if(largest != i)
        {
            Swap(i, largest);
            Modify(largest);
        }
    }
    private void Swap(int i, int j)
    {
        int tmp = elements[i];
        elements[i] = elements[j];
        elements[j] = tmp;
    }
    public void Display()
    {
        for (int i = 0; i < Count; i++)
        {
            Console.Write(" " + elements[i]);
        }
        Console.WriteLine();
    }
}