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.Linq;
using System.Text;
using System.Threading.Tasks;

namespace lec09B
{
    class Program
    {
        static int size;

        static void Main(string[] args)
        {
            //setting part
            size = int.Parse(Console.ReadLine());
            int[] array = new int[size];
            string input = Console.ReadLine();
            string[] split = input.Split(' ');

            for (int i = 0; i < size; i++)
            {
                array[i] = int.Parse(split[i]);
            }

            //heap part
            BuildMaxHeap(array);

            //printf part
            foreach (int element in array){
                Console.Write(" {0}",element);
            }

            Console.WriteLine();
            Console.ReadLine();
        }

        static void BuildMaxHeap(int [] arr )
        {
            for (int i = size / 2 ; i > 0; i--)
            {
                MaxHeapify(arr, i);
            }
        }

        static void MaxHeapify (int [] arr , int i)
        {
            int l = i * 2;
            int r = i * 2 + 1;
            int largest;

            if ( l <= size && arr[l-1] > arr[i-1])
            {
                largest = l;
            } else
            {
                largest = i;
            }
            if ( r <= size && arr[r-1] > arr[largest-1])
            {
                largest = r;
            } 

            if (largest != i)
            {
                int temp = arr[i-1];
                arr[i-1] = arr[largest-1];
                arr[largest-1] = temp;

                MaxHeapify(arr , largest);
            }
        }
    }
}