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 Scala (judge verdict: Wrong Answer)

import scala.io.StdIn.{readInt, readLine};

object Main {
    def main(args: Array[String]): Unit = {
        val n = readInt;
        val arr = readLine.split(" ").map(_.toInt);
        build(n, arr);
        println(" " + arr.mkString(" "));
    }
    
    def build(length: Int, arr: Array[Int]): Unit = {
        def go(curr: Int): Unit = {
            if (curr < length) {
                go(curr * 2 + 1);
                heapify(length, arr, curr);
                go(curr * 2 + 2);
            }
        }
        go(0);
    }
    
    def heapify(length: Int, arr: Array[Int], idx: Int): Unit = {
        val left = idx * 2 + 1;
        val right = idx * 2 + 2;
        if (right < length) {
            val m = max(arr, idx, left, right);
            if (idx != m) {
                swap(arr, idx, m);
                heapify(length, arr, m);
            }
        } else if (left < length) {
            if (arr(left) > arr(idx)) swap(arr, left, idx);
        } else {
            // nop
        }
    }
    
    def max(arr: Array[Int], k1: Int, k2: Int, k3: Int): Int = {
        val v1 = arr(k1); val v2 = arr(k2); val v3 = arr(k3);
        if (v1 > v2) {
            if (v1 > v3) k1 else k3;
        } else {
            if (v2 > v3) k2 else k3;
        }
    }
    
    def swap(arr: Array[Int], a: Int, b: Int): Unit = {
        val tmp = arr(a);
        arr(a) = arr(b);
        arr(b) = tmp;
    }
}