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;

namespace ALDS1_9_B {
    class Problem {
        private int _H;
        private int[] _A;

        public void Run() {
            Read();

            for (int i = _H / 2; i > 0; i--)
                MaxHeapify(i);

            StringBuilder sb = new StringBuilder();
            for (int i = 1; i <= _H; i++)
                sb.Append(" " + _A[i]);
            Console.WriteLine(sb);
        }

        private void Read() {
            _H = int.Parse(Console.ReadLine());
            _A = new int[_H + 1];
            string[] input = Console.ReadLine().Split(' ');
            for (int i = 1; i <= _H; i++)
                _A[i] = int.Parse(input[i - 1]);
        }

        private void MaxHeapify(int i) {
            int l = 2 * i;
            int r = 2 * i + 1;
            int largest = (l <= _H && _A[l] > _A[i]) ? l : i;
            if (r <= _H && _A[r] > _A[largest])
                largest = r;

            if (largest != i) {
                Swap(i, largest);
                MaxHeapify(largest);
            }
        }

        private void Swap(int a,int b) {
            int temp = _A[a];
            _A[a] = _A[b];
            _A[b] = temp;
        }
    }
    
    class Program {
        static void Main(string[] args) {
            Problem p = new Problem();
            p.Run();
        }
    }
}