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 JAVA (judge verdict: Wrong Answer)


import java.lang.reflect.Array;
import java.util.Comparator;
import java.util.Scanner;

public class Main {

    public static class PriorityQueue<T> {

        private Comparator<T> comparator;
        private T[] array;
        private int lastIdx = 0;
        private Class clazz;

        public PriorityQueue(Comparator<T> comparator, Class clazz) {
            this(comparator, clazz, 50);
        }

        public PriorityQueue(Comparator<T> comparator, T[] array) {
            this.comparator = comparator;
            if (array.length == 0 || array[1] == null) {
                throw new IllegalArgumentException("array should have at least one element at index 1");
            }
            this.clazz = array[1].getClass();
            this.array = array;
            heapify();
        }

        private void heapify() {
            lastIdx = array.length - 1;
            for (int i = array.length / 2; i >= 1; i--) {
                sink(i);
            }
        }

        @SuppressWarnings("unchecked")
        public PriorityQueue(Comparator<T> comparator, Class clazz, int initialCapacity) {
            this.comparator = comparator;
            this.clazz = clazz;
            array = (T[]) Array.newInstance(clazz, initialCapacity);
        }

        private void swim(int k) {
            while (k > 1 && less(k / 2, k)) {
                exch(k / 2, k);
                k = k / 2;
            }
        }

        private void sink(int k) {
            while (2 * k <= lastIdx) {
                int largerChild = 2 * k;
                if (largerChild + 1 <= lastIdx && less(largerChild, largerChild + 1)) {
                    largerChild++;
                }
                if (!less(k, largerChild)) {
                    break;
                }
                exch(k, largerChild);
                k = largerChild;
            }
        }

        private void exch(int i, int j) {
            T key = array[i];
            array[i] = array[j];
            array[j] = key;
        }

        private boolean less(int i, int j) {
            return comparator.compare(array[i], array[j]) < 0;
        }

        public T peek() {
            return lastIdx == 0 ? null : array[1];
        }

        public T pop() {
            if (lastIdx == 0) {
                return null;
            }
            exch(1, lastIdx);
            T top = array[lastIdx--];
            sink(1);
            return top;
        }

        @SuppressWarnings("unchecked")
        public void insert(T key) {
            if (lastIdx == array.length - 1) {
                T[] newArray = (T[]) Array.newInstance(clazz, array.length * 2);
                for (int i = 1; i < array.length; i++) {
                    newArray[i] = array[i];
                }
                array = newArray;
            }
            array[++lastIdx] = key;
            swim(lastIdx);
        }

        public boolean isEmpty() {
            return lastIdx == 0;
        }

        public int size() {
            return lastIdx;
        }
    }

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int n = in.nextInt();
        Integer[] array = new Integer[n + 1];
        for (int i = 1; i <= n; i++) {
            array[i] = in.nextInt();
        }
        PriorityQueue<Integer> maxPQ = new PriorityQueue<Integer>(
                new Comparator<Integer>() {
            @Override
            public int compare(Integer o1, Integer o2) {
                return o1 - o2;
            }
        }, array);
        for (int i = 1; i <= n; i++) {
            System.out.println(array[i]);
        }
    }
}