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)

#include<stdio.h>

#define SIZE 500000

int heap[SIZE];

void maxHeapify(int i, int n){
	int l = 2*i;
	int r = 2*i+1;
	int largest = i;
	if( l <= n && heap[l] > heap[i] ){
		largest = l;
	}
	if( r <= n && heap[r] > heap[largest] ){
		largest = r;
	}
	if( largest != i ){
		int temp = heap[i];
		heap[i] = heap[largest];
		heap[largest] = temp;
		maxHeapify(largest, n);
	}
	return;
}

void buildMaxHeap(int n){
	int i;
	for( i = SIZE/2; i >= 1; i-- ){
		maxHeapify(i, n);
	}
}

void output(int k, int n){
	printf("node %d: key = %d, ", k, heap[k]);
	if( k/2 > 0 ){ printf("parent key = %d, ", heap[k/2]); }
	if( k*2 <= n ){ printf("left key = %d, ", heap[k*2]); }
	if( k*2 + 1 <= n ){ printf("right key = %d, ", heap[k*2+1]); }
	printf("\n");
	return;
}

int main(){
	int n, i;
	scanf("%d", &n);
	for( i = 1; i <= n; i++ ){ scanf("%d", &heap[i]); }
	buildMaxHeap(n);
	for( i = 1; i <= n; i++ ){ printf(" %d", heap[i]); }
	printf("\n");
	return 0;
}