Slowsort

Slowsort is a recursive algorithm.

An in-place implementation in pseudo code:

procedure slowsort(A, i, j)                           // sorts Array A[i],...,A[j]
    if i ≥ j then
        return
    m := ⌊(i+j) / 2⌋                            
    slowsort(A, i, m)                                 // (1.1)
    slowsort(A, m+1, j)                               // (1.2)
    if A[j] < A[m] then
        swap A[j] and A[m]                            // (1.3)
    slowsort(A, i, j-1)                               // (2)

• Sort the first half recursively (1.1)
• Sort the second half recursively (1.2)
• Find the maximum of the whole array by comparing the results of 1.1 and 1.2 and place it at the end of the list (1.3)
• Recursively sort the entire list without the maximum in 1.3 (2).

An implementation in Haskell (purely functional) may look as follows.

slowsort :: (Ord a) => [a] -> [a]
slowsort xs
  | length xs <= 1 = xs
  | otherwise      = slowsort xs' ++ [max llast rlast]  -- (2)
  where m     = length xs `div` 2
        l     = slowsort $ take m xs  -- (1.1)
        r     = slowsort $ drop m xs  -- (1.2)
        llast = last l
        rlast = last r
        xs'   = init l ++ min llast rlast : init r

The runtime ${\displaystyle T(n)}$ for Slowsort is ${\displaystyle T(n)=2T(n/2)+T(n-1)+1}$. A lower asymptotic bound for ${\displaystyle T(n)}$ in Landau notation is ${\displaystyle \Omega \left(n^{\frac {\log _{2}(n)}{(2+\epsilon )}}\right)}$ for any ${\displaystyle \epsilon >0}$. Slowsort is therefore not in polynomial time. Even the best case is worse than Bubble sort.