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 PHP (judge verdict: WA: Presentation Error)

<?php
	function swap(&$array, $i, $j){
		$tmp = $array[$i];
		$array[$i] = $array[$j];
		$array[$j] = $tmp;
	}

	//$array[$node]をヒープの正しい位置に来るようにする関数
	function max_heapify(&$array, $node, $heap_size){
		$left = 2 * $node + 1;	//配列$arrayの添え字 $nodeの左の子を示す添え字
		$right = 2 * $node + 2;	//$nodeの右のノード $nodeの右の子を示す添え字
		$largest;	//$node $left $right の中で配列の要素が最も大きいものの添え字

		//
		if($left < $heap_size && $array[$left] > $array[$node])
			$largest = $left;
		else
			$largest = $node;
		if($right < $heap_size && $array[$right] > $array[$largest])
			$largest = $right;

		if($largest != $node){
			swap($array, $largest, $node);
			max_heapify($array, $largest, $heap_size);
		}
	}

	function build_max_heap(&$array){
		for($i = (int)(count($array) / 2 - 1); $i >= 0; --$i)
			max_heapify($array, $i, count($array));
	}

	function heapSort(&$array){
		$heap_size = count($array);
		build_max_heap($array);
		for($i = count($array) - 1; $i >=0; --$i){
			swap($array, $i, 0);
			--$heap_size;
			max_heapify($array, 0, $heap_size);
		}
	}

	$n = intval(trim(fgets(STDIN)));
	$string = trim(fgets(STDIN));
	$numbers = explode(" ", $string);

	build_max_heap($numbers);

	for($i = 0; $i < count($numbers); ++$i){
		echo $numbers[$i];
		if($i != (count($numbers) - 1))
			echo " ";
	}
	echo PHP_EOL;
?>