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<bits/stdc++.h>
using namespace std;

vector<int> heap;
int hp = 1;

// void propagate(int i)
// {
//   if (i == 1) return;
//   int parent = i/2;
//   if (heap[parent] < heap[i]) {
//     swap(heap[i], heap[parent]);
//     propagate(parent);
//   }
// }
// 
// void insert(int v)
// {
//   heap[hp] = v;
//   propagate(hp);
//   hp++;
// }

void max_heapify(int n, int i)
{
  int li = 2*i;
  int ri = 2*i+1;

  int largest;
  if (li <= n && heap[li] > heap[i])
    largest = li;
  else
    largest = i;
  if (ri <= n && heap[ri] > heap[largest])
    largest = ri;

  if (largest != i) {
    swap(heap[i], heap[largest]);
    max_heapify(n, largest);
  }
}

int main()
{
  int n; cin >> n;
  heap = vector<int>(n+1);
  for (int i = 0; i < n; i++) {
    int v; cin >> v;
    // insert(v);
    heap[hp++] = v;
  }
  for (int i = n; i >= 1; i--)
    max_heapify(n, i);
  for (int i = 1; i <= n; i++)
    cout << " " << heap[i];
  cout << endl;
  return EXIT_SUCCESS;
}