Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

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>
#include <stdbool.h>

#define SWAP(type, a, b) do { type t = a; a = b; b = t; } while(0)

typedef struct {
    int key;
} Node;

typedef struct {
    Node *node;
    int size;
} Heap;

int getParent(int idx);
int getLeft(int idx);
int getRight(int idx);
bool isCorrectHeapIdx(int idx, int heapSize);
bool initialize(Heap *heap, int size);
void maxHeapify(Heap *heap, int idx);
void buildMaxHeap(Heap *heap);
void printHeap(Heap *heap);
void terminate(Heap *heap);

int getParent(int idx) {
    return idx / 2;
}

int getLeft(int idx) {
    return idx * 2;
}

int getRight(int idx) {
    return idx * 2 + 1;
}

bool isCorrectHeapIdx(int idx, int heapSize) {
    return idx <= heapSize && 1 <= idx;
}

bool initialize(Heap *heap, int size) {
    if((heap->node = malloc((size + 1) * sizeof(Node))) == NULL) {
        return false;
    }

    heap->size = size;
    return true;
}

void maxHeapify(Heap *heap, int idx) {
    int left = getLeft(idx);
    int right = getRight(idx);

    int maxIdx = idx;
    if(isCorrectHeapIdx(left, heap->size) && heap->node[maxIdx].key < heap->node[left].key) {
        maxIdx = left;
    } if(isCorrectHeapIdx(right, heap->size) && heap->node[maxIdx].key < heap->node[right].key) {
        maxIdx = right;
    }

    if(maxIdx != idx) {
        SWAP(int, heap->node[maxIdx].key, heap->node[idx].key);
        maxHeapify(heap, maxIdx);
    }
}

void buildMaxHeap(Heap *heap) {
    for(int i = getParent(heap->size); i >= 1; i--) {
        maxHeapify(heap, i);
    }
}

void printHeap(Heap *heap) {
    for(int i = 1; i <= heap->size; i++) {
        printf(" %d", heap->node[i].key);        
    }
    puts("");
}

void terminate(Heap *heap) {
    free(heap->node);
    heap->node = NULL;
    heap->size = 0;
}

int main(void) {
    int n;
    Heap heap;

    scanf("%d", &n);
    initialize(&heap, n);

    for(int i = 1; i <= n; i++) {
        scanf("%d", &heap.node[i].key);
    }

    buildMaxHeap(&heap);

    printHeap(&heap);

    terminate(&heap);
    return 0;
}