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: Runtime Error)

<?php
solve1();

// 自力解
function solve1(){
	$h = (int)trim(fgets(STDIN));
	$input = explode(' ', trim(fgets(STDIN)));
	$data = [];
	for ($i=0; $i < $h; $i++) { 
		$data[$i+1] = (int)$input[$i];
	}
	$start = (int)floor($h/2);
	for ($i=$start; $i > 0; $i--) { 
		$data = maxHeapify($data, $i);
	}
	foreach ($data as $v) {
		echo ' ' . $v;
	}
	echo PHP_EOL;
}

function maxHeapify($array, $i){
	$h = count($array);
	$l_idx = 2 * $i;
	$r_idx = (2 * $i) + 1;
	$largest = $i;
	if ($l_idx <= $h && $array[$l_idx] > $array[$i]) {
		$largest = $l_idx;
	}
	if ($r_idx <= $h && $array[$r_idx] > $array[$largest]) {
		$largest = $r_idx;
	}

	if ($largest !== $i) {
		$tmp = $array[$largest];
		$array[$largest] = $array[$i];
		$array[$i] = $tmp;
		$array = maxHeapify($array, $largest);
	}
	return $array;
}