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 MAX_HEAP_SIZE   500001      /* 最大ヒープ領域(ヒープ配列を1オリジンで持つため500000+1) */

int HeapSize;
int HeapArray[MAX_HEAP_SIZE];       /* ヒープ配列(1オリジン) */

void
initVer( void )
{
    int i;
    
    for( i = 0; i < MAX_HEAP_SIZE; i++ )
    {
        HeapArray[i] = 0;
    }
}

void
inputVal( void )
{
    int i;
    
    scanf( "%d", &HeapSize );
    /* 要素数1から入力 */
    for( i = 1; i <= HeapSize; i++ )
    {
        scanf( "%d", &HeapArray[i]);
    }
}

/****************************************************************************/
/* 関数名   | maxHeapify                                                    */
/* 関数仕様 | Nodeを根とする部分木をmax-ヒープ木にする                      */
/****************************************************************************/
void
maxHeapify( int Node )
{
    int LeftNode;
    int RightNode;
    int LargetNode;
    int TmpHeapVal;
    
    /* ヒープ配列は、Nodeの要素数をnとすると、Nodeの子の要素数は左が2n、右が2n×1 */
    LeftNode = 2 * Node;
    RightNode = LeftNode + 1;
    
    /* NodeとNodeの子の中で、最大値の要素数を求める */
    if( ( LeftNode <= HeapSize )
    &&  ( HeapArray[LeftNode] > HeapArray[Node] ) )
    {
        LargetNode = LeftNode;
    }else{
        LargetNode = Node;
    }
    if( ( RightNode <= HeapSize )
    &&  ( HeapArray[RightNode] > HeapArray[LargetNode] ) )
    {
        LargetNode = RightNode;
    }
    
    /* Nodeに対しNodeの子の方が大きい場合、NodeとNodeの子の値を入れ替える */
    if( LargetNode != Node )
    {
        TmpHeapVal = HeapArray[Node];
        HeapArray[Node] = HeapArray[LargetNode];
        HeapArray[LargetNode] = TmpHeapVal;
        /* NodeとNodeの子の値を入れ替えたヒープ配列に対し、Nodeの子にmaxHeapifyを適用する */
        maxHeapify( LargetNode );
    }
}

/****************************************************************************/
/* 関数名   | buildMaxHeap                                                  */
/* 関数仕様 | 子を持つ節に対しボトムアップにmaxHeapifyを適用し              */
/*          | 配列をmax-ヒープに変換する                                    */
/****************************************************************************/
void buildMaxHeap(void){
    int i;
    
    /* 子を持つ節の中で最大の要素数がHeapSize/2 */
    /* 要素数がHeapSize/2節からボトムアップにmaxHeapifyを適用する */
    for( i = HeapSize / 2; i > 0; i-- )
    {
        maxHeapify( i );
    }
}

void outputVal(void){
    int i;
    /* 要素数1から出力 */
    for( i = 1; i <= HeapSize; i++ )
    {
        printf( " %d", HeapArray[i] );
    }
    printf( "\n" );
}

int main(void){
    initVer();
    /* 入力部 */
    inputVal();
    
    /* 制御部 */
    buildMaxHeap();
    
    /* 出力部 */
    outputVal();
}