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 HeapTree
	def initialize
		@node = []
	end

	private

	def swap(key)
		if  key < 0 
			return nil
		end
		if !left(key) && !right(key)
			return nil
		elsif left(key) && !right(key)
			if left(key) > @node[key]
				@node[key], @node[left_key(key)] =  @node[left_key(key)], @node[key]
				swap(left_key(key))
			end
		elsif right(key) && !left(key)
			if right(key) > @node[key]
				@node[key], @node[right_key(key)] =  @node[right_key(key)], @node[key]
					swap(right_key(key))
			end
		elsif @node[key] < left(key) && left(key) > right(key)
			@node[key], @node[left_key(key)] =  @node[left_key(key)], @node[key]
				swap(left_key(key))
		elsif @node[key] < right(key) && right(key) > left(key)
			@node[key], @node[right_key(key)] =  @node[right_key(key)], @node[key]
				swap(right_key(key))
		end
	end

	def extract!
		num = @node[0]
		@node[0] = @node.pop
		swap(@node[@node.length-1])
		return num
	end

	def insert!(num)
		@node << num
		swap(@node.length.to_i - 1)
	end

	def left(key)
		@node[2*key+1]
	end

	def right(key)
		@node[2*key+2]
	end

	def left_key(key)
		2*key+1
	end

	def right_key(key)
		2*key+2
	end

	# ???????????¢?´¢
	def precedingOrderSearch(key, &func)
		if @node[key]
			func.call(key, @node[key])
			precedingOrderSearch(left(key), &func)
			precedingOrderSearch(right(key), &func)
		end
	end
	# ???????????¢?´¢
	def afterLineOrderSearch(key, &func)
		if @node[key]
			afterLineOrderSearch(left(key), &func)
			afterLineOrderSearch(right(key), &func)
			func.call(key, @node[key])
		end
	end
	#???????????¢?´¢
	def centerOrderSearch(node, &func)
		if @node[key]
			centerOrderSearch(left(key), &func)
			func.call(key, @node[key])
			centerOrderSearch(right(key), &func)
		end
	end

	public

	def insert(num)
		insert!(num)
	end

	def preceding_crawl(&func) #???hoge.precedingCrawl do |key, value| ??????????????????
		precedingOrderSearch(0, &func)
	end

 # ????????????
 def after_line_crawl(&func)
 	afterLineOrderSearch(0, &func)
 end

# ????????????
def center_crawl(&func)
	centerOrderSearch(0, &func)
end

def extract
	extract!
end

def priority_create
	i = 0
loop{
	key = (@node.size.to_i-1)/2-i
	break if key < 0
	swap(key)
	i += 1
}
end

def list
	return @node
end
end


head = HeapTree.new
n = gets.to_i
 gets.split.map(&:to_i).each do |v|
 	head.insert v
 end
head.priority_create
head.list.each do |v|
	print " #{v}"
end
print "\n"