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>
#include <stdlib.h>

#define MAX_HEAP_LENGTH 100

void swap(int array[], int i, int j) {
    int temp = array[i];
    array[i] = array[j];
    array[j] = temp;
}

typedef struct {
    int * array;
    int heap_size;
    int length;
} Heap;

Heap createHeap(int length) {
    Heap heap;
    
    heap.array = (int *)malloc((length+1) * sizeof(int));
    heap.array[0] = 0; // NOT USED
    heap.heap_size = heap.length = length;
    
    return heap;
}

int parentOfIndex(int index) {
    return index/2;
}
int leftOfIndex(int index) {
    return 2*index;
}
int rightOfIndex(int index) {
    return 2*index+1;
}

void maxHeapify(Heap heap, int iThis) {
    int iLeft = leftOfIndex(iThis);
    int iRight = rightOfIndex(iThis);
    int iLargest;
    
//    printf("HeapMaxify: L:%d,R:%d,This:%d\n", 
//            heap.array[iLeft],
//            heap.array[iRight],
//            heap.array[iThis]
//            );
    
    if (iLeft <= heap.heap_size && heap.array[iLeft] > heap.array[iThis]) 
        iLargest = iLeft;
    else
        iLargest = iThis;
    
    if (iRight <= heap.heap_size && heap.array[iRight] > heap.array[iLargest]) 
        iLargest = iRight;
    
    if (iLargest != iThis) {
        swap(heap.array, iThis, iLargest);
        // iLargest is no longer the index for largest value within the trinode.
        maxHeapify(heap, iLargest);
    }
}
void buildMaxHeap(Heap heap) {
    heap.heap_size = heap.length;
    
    for (int i = heap.length/2; i >= 1; i--)
        maxHeapify(heap, i);
}
void heapSort(Heap heap) {
    buildMaxHeap(heap);
    
    for (int i = heap.length; i >= 2; i--) {
        swap(heap.array, 1, i);
        heap.heap_size--;
        maxHeapify(heap, 1);
    }
}


int main(int argc, char** argv) {
    int nNode;
    scanf("%d", &nNode);
    
    Heap heap = createHeap(nNode);
    // 1-origin
    
    for (int i = 1; i <= nNode; i++)
        scanf("%d", &(heap.array[i]));
    
    buildMaxHeap(heap);
    
    for (int i = 1; i <= heap.length; i++) 
        printf(" %d", heap.array[i]);
    
    printf("\n");
    
//    for (int i = 1; i <= nNode; i++) {
//        printf("node %d: ", i);
//        printf("key = %d, ", heap.array[i]);
//        
//        if (i/2 > 0)
//            printf("parent key = %d, ", heap.array[i/2]);
//        
//        if (2*i <= heap.length)
//            printf("left key = %d, ", heap.array[2*i]);
//        
//        if (2*i+1 <= heap.length)
//            printf("right key = %d, ", heap.array[2*i+1]);
//        
//        printf("\n");
//    }

    return (EXIT_SUCCESS);
}