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.Text;
					
public class Program
{
	public static void Main()
	{
		int n = int.Parse(Console.ReadLine());
		
		int[] heap = Array.ConvertAll(Console.ReadLine().Split(" ".ToCharArray()), int.Parse);
		
		BuildMaxHeap(heap);
		
		StringBuilder sb = new StringBuilder();
		for(int i = 0; i < n; i ++)  {
			sb.Append(" " + heap[i]);
		}
		Console.WriteLine(sb);
	}
	
	public static string ConvertString(int[] heap, int i)
	{
		string result = "node " + (i + 1) ;
		
		result += ": key = " + heap[i];
		
		if(i > 0) {
			result += ", parent key = " + heap[(i-1)/2];
		}
		
		if( 2*i+1 < heap.Length) {
			result += ", left key = " + heap[2*i+1];
		}
		
		if( 2 * (i+1) < heap.Length) {
			result += ", right key = " + heap[2*i + 2];
		}
		
		return result + ", ";
	}
	
	public static void BuildMaxHeap(int[] A)
	{
		for(int i = A.Length / 2; i >= 0; i--){
			MaxHeapify(A, i);
		}
	}
	
	public static void MaxHeapify(int[] A, int i)
	{
		int l = Left(A, i);
		int r = Right(A, i);
		int largest = i;
		
		//l,r,i?£???????????£???§???????\???§?£???????????£???¶?£????
		if(l >= 0) {
			if(A[l] > A[largest]) largest = l;	
		}
		if(r >= 0) {
			if(A[r] > A[largest]) largest = r;
		}
		
		if(i != largest) {
			Swap(A, i, largest);
			MaxHeapify(A, largest);
		}
	}
	
	public static int Left(int[] A, int i)
	{	
		return ( (2*i + 1) < A.Length )? (2*i + 1) : -1;
	}
	
	public static int Right(int[] A, int i)
	{	
		return ( 2 * (i+1) < A.Length )? (2*i + 2) : -1;
	}
	
	public static void Swap(int[] A, int i, int j)
	{
		int tmp = A[i];
		A[i] = A[j];
		A[j] = tmp;
	}
}