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: Accepted)

class Maximum
  def initialize
    @arr = []
    monky
  end
  def monky
    def @arr.[](i)
      super(i - 1)
    end
    def @arr.[]=(i, value)
      super(i - 1, value)
    end
  end
  def buildHeap(root = 1)
    left, right = root * 2, root * 2 + 1
    if @arr[right]
      buildHeap(left); buildHeap(right)
      max = [[@arr[left], left], [@arr[right], right]].max
      if max[0] > @arr[root]
        @arr[root], @arr[max[1]] = @arr[max[1]], @arr[root]
        rebuildHeap(max[1])
      end
    elsif @arr[left]
      buildHeap(left)
      if @arr[left] > @arr[root]
        @arr[root], @arr[left] = @arr[left], @arr[root]
        rebuildHeap(left)
      end                                                       
    end
  end
  def rebuildHeap(root = 1)
    left, right = root * 2, root * 2 + 1
    if @arr[right]
      max = [[@arr[left], left], [@arr[right], right]].max
      if max[0] > @arr[root]
        @arr[root], @arr[max[1]] = @arr[max[1]], @arr[root]
        rebuildHeap(max[1])
      end
    elsif @arr[left]
      if @arr[left] > @arr[root]
        @arr[root], @arr[left] = @arr[left], @arr[root]
        rebuildHeap(left)
      end
    end      
  end
  def insert(array)
    @arr = array
    monky
    buildHeap
  end
  def to_s
    @arr.join(" ")
  end
end
def main(input = STDIN)
  heap = Maximum.new
  input.gets
  heap.insert(input.gets.split(" ").map(&:to_i))
  puts " " + heap.to_s
end
main