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.Collections.Generic;
using System.Text;

class Program
{

    public void Proc()
    {
        int itemCount = int.Parse(Console.ReadLine());
        string[] inpt = Console.ReadLine().Split(' ');
        long[] arr = new long[itemCount];


        for (int i = 0; i < itemCount; i++)
        {
            long target = long.Parse(inpt[i]);
            arr[i] = target;
        }

        for (int i = itemCount / 2; i >= 0; i--)
        {
            MaxHieapfy(arr, i);
        }
        for (int i = 0; i < arr.Length; i++)
        {
            Console.Write(" " + arr[i]);
        }
        Console.WriteLine("");
    }

    public void MaxHieapfy(long[] arr, int rootIdx)
    {
        // ????????????INDEX
        int leftChildIndex = this.GetLeftChildIndex(rootIdx);
        int rightChildIndex = leftChildIndex + 1;
        int largest = -1;
        if (leftChildIndex < arr.Length && arr[rootIdx] < arr[leftChildIndex])
        {
            largest = leftChildIndex;
        }
        else
        {
            largest = rootIdx;
        }
        if (rightChildIndex < arr.Length && arr[largest] < arr[rightChildIndex])
        {
            largest = rightChildIndex;
        }
        if (largest != rootIdx)
        {
            long tmp = arr[rootIdx];
            arr[rootIdx] = arr[largest];
            arr[largest] = tmp;
            this.MaxHieapfy(arr, largest);
        }
    }


    private Dictionary<int, int> LeftChildDic = new Dictionary<int, int>();
    private int GetLeftChildIndex(int parentIndex)
    {
        if (LeftChildDic.ContainsKey(parentIndex))
        {
            return LeftChildDic[parentIndex];
        }
        int prev = 0;
        int idx = 0;
        for (int i = 0; i <= parentIndex; i++)
        {
            int plus = (int)Math.Pow(2, i);
            if (parentIndex >= prev && parentIndex < prev + plus)
            {
                int innerIdx = parentIndex - prev;
                idx = (innerIdx * 2) + (prev + plus);
                this.LeftChildDic.Add(parentIndex, idx);
                break;
            }
            else
            {
                prev += plus;
            }
        }
        return idx;
    }

    static void Main(string[] args)
    {
        Program prg = new Program();
        prg.Proc();
    }
}