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: Time Limit Exceeded)

#include<stdio.h>

#define MAX 250

struct node{
  int key;
};

int h;

int Parent(int i){
  if(i == 1) return 0;
  return i / 2;
}

int Left(int i){
  if(h < (2 * i)) return 0;
  else return 2 * i;
}

int Right(int i){
  if(h < (2 * i + 1)) return 0;
  return (2 * i) + 1;
}

//void problem1(struct node *n){
//  int i;
//  for(i = 1; i <= h; i++){
//    printf("node %d: key = %d, ",i,n[i].key);
//    if(Parent(i) != 0)
//      printf("parent key = %d, ",n[Parent(i)].key);
//    if(Left(i) != 0)
//      printf("left key = %d, ",n[Left(i)].key);
//    if(Right(i) != 0)
//      printf("right key = %d, ",n[Right(i)].key);
//    printf("\n");
//  }
//}

void maxHeapify(struct node *A, int i){
  int l,r,largest,buf;

  l = Left(i);
  r = Right(i);

  if(l <= MAX && A[l].key > A[i].key)
    largest = l;
  else
    largest = i;

  if(r <= MAX && A[r].key > A[largest].key)
    largest = r;

  if(largest != i){
    buf = A[i].key;
    A[i].key = A[largest].key;
    A[largest].key = buf;
    maxHeapify(A,largest);
  }
    
}

void buildMaxHeap(struct node *A){
  int i;
  for(i = h / 2; i > 0; i--)
    maxHeapify(A,i);
}

int main(){
  int i;
  struct node n[MAX];

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

  buildMaxHeap(n);
  for(i = 1; i <= h; i++)
    printf("%d ",n[i].key);
  printf("\n");
  
  return 0;
}