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: Accepted)

import scala.io.StdIn.{readInt, readLine};
import scala.reflect.ClassTag;
object Main {
    def main(args: Array[String]): Unit = {
        val n = readInt; // 要素数
        val arr = readLine.split(" ").map(_.toLong);
        val h = Heap[Long](arr, (a: Long, b: Long) => if (a < b) -1 else if (a > b) 1 else 0);
        h.dump;
    }
}
class Heap[A](heap: Array[Option[A]], compare: (A, A) => Int) {
    val size = heap.length;
    var ptr = 1;
    def dump(): Unit = {
        println(" " + heap.filter(_ != None).map(_.get).mkString(" "));
    }
    def add(key: A): Unit = {
        if (ptr < size) {
            heap(ptr) = Some(key);
            buildHeap();
            ptr += 1;
        }
    }
    
    private def buildHeap(): Unit = {
        def _go(idx: Int): Unit = {
            if (idx < size && heap(idx) != None) {
                _go(idx * 2 + 1);
                _go(idx * 2 + 2);
                heapify(idx)
            }
        }
        _go(0);
    }
    
    private def heapify(idx: Int): Unit = {
        val lIdx = idx * 2 + 1;
        val rIdx = idx * 2 + 2;
        val curr = heap(idx);
        val left = if (lIdx < size) heap(lIdx) else None;
        val right = if (rIdx < size) heap(rIdx) else None;
        if (left != None && right != None) { // 左右の子ノードが存在
            if (compare(curr.get, left.get) < 0 || compare(curr.get, right.get) < 0 ) {
                val sp = if (compare(left.get, right.get) < 0) rIdx else lIdx;
                swap(idx, sp);
                heapify(sp)
            }
        } else if (left != None) { // 左の子ノードのみ存在
            if (compare(curr.get, left.get) < 0) {
                swap(idx, lIdx);
                heapify(lIdx)
            }
        }
    }

    private def swap(idx1: Int, idx2: Int): Unit = {
        val tmp = heap(idx1);
        heap(idx1) = heap(idx2);
        heap(idx2) = tmp;
    }
}

object Heap {
    def apply[A](arr: Array[A], compare: (A, A) => Int): Heap[A] = {
        val h = new Heap[A](arr.map(Option[A]), compare);
        h.buildHeap();
        h
    }
    def apply[A](size: Int, compare: (A, A) => Int)(implicit m: ClassTag[Option[A]]): Heap[A] = {
        val arr = m.newArray(size);
        new Heap[A](arr, compare);
    }
}