heapsort input with most and fewest comparisons

Question

Hi I've looked around a bit and haven't been able to find any direct discussion of this question. Most seem to cover time complexity and the big O notation.

I'm wondering if and how the order of input into a heapsort algorithm will impact the number of comparisons needed to sort the input. For example, take a heapsort algorithm that sorts in ascending order (smallest to largest)....if I input a set of integers already ordered in this way (ascending) how many comparisons would it require compared to a set of input that is ordered in a descending manner (largest to smallest)? How about compared to a completely randomized set of the same numbers?

public class Heap {
    // This class should not be instantiated.
    private Heap() {
    }

    /**
     * Rearranges the array in ascending order, using the natural order.
     * 
     * @param pq
     *            the array to be sorted
     */
    public static void sort(Comparable[] pq) {
        int N = pq.length;
        for (int k = N / 2; k >= 1; k--)
            sink(pq, k, N);
        while (N > 1) {
            exch(pq, 1, N--);
            sink(pq, 1, N);
        }
    }

    private static void sink(Comparable[] pq, int k, int N) {
        while (2 * k <= N) {
            int j = 2 * k;
            if (j < N && less(pq, j, j + 1))
                j++;
            if (!less(pq, k, j))
                break;
            exch(pq, k, j);
            k = j;
        }
    }

    private static boolean less(Comparable[] pq, int i, int j) {
        return pq[i - 1].compareTo(pq[j - 1]) < 0;
    }

    private static void exch(Object[] pq, int i, int j) {
        Object swap = pq[i - 1];
        pq[i - 1] = pq[j - 1];
        pq[j - 1] = swap;
    }

    // is v < w ?
    private static boolean less(Comparable v, Comparable w) {
        return (v.compareTo(w) < 0);
    }
}

Answer 1

Heap sort is different from bubble sort and quick sort, the best and worst cases wouldn't happen when the input elements are ordered in a descending/ascending manner. The first step of heap sort is building a heap(max-heap in general) which can be done in linear time O(n) by using the "sift down" version of heapify, no matter what order the initial elements are. If the input is already a heap, it just saves the time for exchange.
It's generally considered the best and worst cases performance are both O(nlogn)(the heap sort item on wiki says the best case performance can be Ω(n), and there's a link about it), but the big-O notation omits constant factors and lower order terms, so they are equivalent in big O notation, but they can differ by a constant factor.

For example, I get all 720 permutations of a given elemnts:{1,2,3,4,5,6} and sort them with your code, the minimum/maximum number of comparisons is 12/17 while the order is {6,1,4,2,3,5}/{1,4,2,5,6,3} respectively. And the minimum/maximum number of exchanges is 8/15 while the order is {6,3,5,1,2,4}/{1,2,3,5,4,6} respectively. My another post is about the maximum number of exchanges.