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 Ruby (judge verdict: Wrong Answer)

class Node
  include Comparable

  def initialize(id, key)
    @id = id
    @key = key
  end

  def <=>(other)
    @key <=> other.key
  end

  DUMMY = Node.new(-1, -1)

  attr_accessor :id, :key
end

def maxHeapify(array, i)
  left = (i*2 < array.size) ? array[i*2] : Node::DUMMY
  right = (i*2+1 < array.size) ? array[i*2+1] : Node::DUMMY
  max = [array[i], left, right].sort_by{|x| x.key}.pop
  max_key = max.key
  if (left.key == max.key) then
    left.key = array[i].key
    array[i].key = max_key
    maxHeapify(array, left.id)
  elsif (right.key == max.key) then
    right.key = array[i].key
    array[i].key = max_key
    maxHeapify(array, right.id)
  end
end

def buildMaxHeap(array)
  ((array.size-1) / 2).downto(1) do |i|
    maxHeapify(array, i)
  end
end


# main
n = gets.chomp.to_i
input = gets.chomp.split.map(&:to_i)
array = [n]
input.each_with_index do |key, id|
  id += 1
  node = Node.new(id, key)
  array << node
end

buildMaxHeap(array)
array.shift
puts " #{array.map{|x| x.key}.join(' ')}"