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.Linq;
using System.Text;

namespace ALDS1_9_A
{
    class Program
    {
        static void Main(string[] args)
        {
            string line;
            int[] vals;

            //入力
            line = Console.ReadLine().Trim();
            var heapSize = int.Parse(line);

            line = Console.ReadLine().Trim();
            vals = line.Split(" ".ToCharArray(), StringSplitOptions.RemoveEmptyEntries).Select(e => int.Parse(e)).ToArray();

            var tree = new CompleteBinaryTree(heapSize);
            for (int i = 0; i < heapSize; i++)
                tree[i + 1] = vals[i];

            tree.BuildMaxHeap();

            Console.WriteLine(" " + String.Join(" ", Enumerable.Range(1, heapSize).Select(i => tree[i]).ToArray()));
        }
    }

    class CompleteBinaryTree
    {
        private readonly int[] heap;
        public CompleteBinaryTree(int heapSize)
        {
            heap = new int[heapSize];
        }

        public int this[int i]
        {
            set { heap[i - 1] = value; }
            get { return heap[i - 1]; }
        }

        private static int Parent(int i) => i / 2;
        private static int Left(int i) => i * 2;
        private static int Right(int i) => i * 2 + 1;

        public void BuildMaxHeap()
        {
            for (int i = heap.Length / 2; i > 0; i--)
            {
                MaxHeapify(i);
            }
        }

        private void MaxHeapify(int i)
        {
            var l = Left(i);
            var r = Right(i);
            var H = heap.Length;
            var largest = (l <= H && this[l] > this[i]) ? l : i;
            if (r <= H && this[r] > this[largest])
            {
                largest = r;
            }
            if (largest != i)
            {
                var dummmy = this[i];
                this[i] = this[largest];
                this[largest] = dummmy;
                MaxHeapify(largest);
            }
        }

        public string ToString(int i)
        {
            var st = new StringBuilder();
            st.Append($"node {i}: key = {this[i]}, ");
            if (Parent(i) > 0)
            {
                st.Append($"parent key = {this[Parent(i)]}, ");
            }
            if (Left(i) <= heap.Length)
            {
                st.Append($"left key = {this[Left(i)]}, ");
            }
            if (Right(i) <= heap.Length)
            {
                st.Append($"right key = {this[Right(i)]}, ");
            }
            return st.ToString();
        }
    }
}