Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

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;

namespace A
{
    class Node
    {
        public int key=0;
        public int parent;
        public int left=0;
        public int right=0;
    }

    class Max{
        public void maxHeapify(ref Node[] node, int i)
        {
            int left = i*2, right = i*2+1, large=i;
            Node change = new Node();
            if (left < node.Length && node[left].key > node[i].key)
            {
                large = left;
            }
            if (right < node.Length && node[right].key > node[large].key)
            {
                large = right;
            }
            if (large != i)
            {
                change = node[large];
                node[large] = node[i];
                node[i] = change;
                maxHeapify(ref node, large);
            }
        }
    }

    class B
    {
        static void Main(string[] args)
        {
            string input1 = Console.ReadLine();
            int n = int.Parse(input1), i;
            Node[] node = new Node[n + 1];
            string[] input2 = new string[n + 1];
            for (i = 0; i <= n; i++)
            {
                node[i] = new Node();
            }
            input2 = Console.ReadLine().Split(' ');
            for (i = 0; i < n; i++)
            {
                node[i + 1].key = int.Parse(input2[i]);
            }

            for (i = 1; i <= n; i++)
            {
                if (i / 2 > 0)
                {
                    node[i].parent = node[i / 2].key;
                }
                if (i * 2 <= n)
                {
                    node[i].left = i * 2;
                }
                if (i * 2 + 1 <= n)
                {
                    node[i].right = 2 * i + 1;
                }
            }

            Max max = new Max();
            for (i = n / 2; i > 0; i--)
            {
                max.maxHeapify(ref node, i);
            }

            for (i = 1; i <= n; i++)
            {
                Console.Write(" " + node[i].key);
            }
            Console.WriteLine();

        }
    }
}