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: WA: Presentation Error)

using System;
using System.Linq;

namespace ALDS1_9_B_MaximumHeap
{
	class Program
	{
		static void Main ( string[] args )
		{
			int dataCount = int.Parse (Console.ReadLine ());
			int[] inputNum = new int[dataCount];
			inputNum = Console.ReadLine ().Split (' ').Select (int.Parse).ToArray ();

			int[] result = Heap.BuildMaxHeap (inputNum);

			string output = string.Join (" ", Array.ConvertAll (result, (num => num.ToString ())));

			Console.WriteLine (output);
		}
	}

	class Heap
	{
		public static int[] BuildMaxHeap ( int[] numArray )
		{
			int[] heapArray = new int[numArray.Length + 1];
			numArray.CopyTo (heapArray, 1);

			for (int lp = heapArray.Length / 2; lp >= 1; lp--)
			{
				MaxHeapify (heapArray, lp);
			}

			return heapArray.Skip (1).ToArray ();
		}

		private static void MaxHeapify ( int[] numArray, int index )
		{
			int myselfIndex = index;
			int leftIndex = LeftKey (index);
			int rightIndex = RightKey (index);

			//??¨?????¨????????§?????§?????????????????¶
			bool isRangeLeft = 1 <= leftIndex && leftIndex < numArray.Length;
			bool isRangeRight = 1 <= rightIndex && rightIndex < numArray.Length;

			int maxIndex = 0;

			if (isRangeLeft && numArray[myselfIndex] < numArray[leftIndex])
			{
				maxIndex = leftIndex;
			}
			else
			{
				maxIndex = myselfIndex;
			}
			if (isRangeRight && numArray[maxIndex] < numArray[rightIndex])
			{
				maxIndex = rightIndex;
			}

			//??????????????§????????´???
			if (maxIndex != index)
			{
				var tmp = numArray[maxIndex];
				numArray[maxIndex] = numArray[index];
				numArray[index] = tmp;

				MaxHeapify (numArray, maxIndex);
			}
		}

		private static int LeftKey ( int index )
		{
			return index * 2;
		}
		private static int RightKey ( int index )
		{
			return index * 2 + 1;
		}
	}
}