Question

I “invented” “new” sort algorithm. Well, I understand that I can’t invent something good, so I tried to search it on wikipedia, but all sort algorithms seems like not my. So I have three questions:

1. What is name of this algorithm?
2. Why it sucks? (best, average and worst time complexity)
3. Can I make it more better still using this idea?

So, idea of my algorithm: if we have an array, we can count number of sorted elements and if this number is less that half of length we can reverse array to make it more sorted. And after that we can sort first half and second half of array. In best case, we need only O(n) – if array is totally sorted in good or bad direction. I have some problems with evaluation of average and worst time complexity.

Code on C#:

    public static void Reverse(int[] array, int begin, int end) {
        int length = end - begin;
        for (int i = 0; i < length / 2; i++)
            Algorithms.Swap(ref array[begin+i], ref array[begin + length - i - 1]);
    }
    public static bool ReverseIf(int[] array, int begin, int end) {
        int countSorted = 1;
        for (int i = begin + 1; i < end; i++)
            if (array[i - 1] <= array[i])
                countSorted++;

        int length = end - begin;
        if (countSorted <= length/2)
            Reverse(array, begin, end);

        if (countSorted == 1 || countSorted == (end - begin))
            return true;
        else
            return false;
    }
    public static void ReverseSort(int[] array, int begin, int end) {
        if (begin == end || begin == end + 1)
            return;
        // if we use if-operator (not while), then array {2,3,1} transforms in array {2,1,3} and algorithm stop
        while (!ReverseIf(array, begin, end)) {
            int pivot = begin + (end - begin) / 2;
            ReverseSort(array, begin, pivot + 1);
            ReverseSort(array, pivot, end);
        }
    }
    public static void ReverseSort(int[] array) {
        ReverseSort(array, 0, array.Length);
    }

P.S.: Sorry for my English.

Answer 1

The best case is Theta(n), for, e.g., a sorted array. The worst case is Theta(n^2 log n).

Upper bound

Secondary subproblems have a sorted array preceded or succeeded by an arbitrary element. These are O(n log n). If preceded, we do O(n) work, solve a secondary subproblem on the first half and then on the second half, and then do O(n) more work – O(n log n). If succeeded, do O(n) work, sort the already sorted first half (O(n)), solve a secondary subproblem on the second half, do O(n) work, solve a secondary subproblem on the first half, sort the already sorted second half (O(n)), do O(n) work – O(n log n).

Now, in the general case, we solve two primary subproblems on the two halves and then slowly exchange elements over the pivot using secondary invocations. There are O(n) exchanges necessary, so a straightforward application of the Master Theorem yields a bound of O(n^2 log n).

Lower bound

For k >= 3, we construct an array A(k) of size 2^k recursively using the above analysis as a guide. The bad cases are the arrays [2^k + 1] + A(k).

Let A(3) = [1, …, 8]. This sorted base case keeps Reverse from being called.

For k > 3, let A(k) = [2^(k-1) + A(k-1)[1], …, 2^(k-1) + A(k-1)[2^(k-1)]] + A(k-1). Note that the primary subproblems of [2^k + 1] + A(k) are equivalent to [2^(k-1) + 1] + A(k-1).

After the primary recursive invocations, the array is [2^(k-1) + 1, …, 2^k, 1, …, 2^(k-1), 2^k + 1]. There are Omega(2^k) elements that have to move Omega(2^k) positions, and each of the secondary invocations that moves an element so far has O(1) sorted subproblems and thus is Omega(n log n).

---

Clearly more coffee is required – the primary subproblems don’t matter. This makes it not too bad to analyze the average case, which is Theta(n^2 log n) as well.

With constant probability, the first half of the array contains at least half of the least quartile and at least half of the greatest quartile. In this case, regardless of whether Reverse happens, there are Omega(n) elements that have to move Omega(n) positions via secondary invocations.