Merge Insertion(Ford-Johnson) Sort in Haskell

MergeInsert.hs

{-
Copyright(C) Freddy A Cubas "Superstar64"

Boost Software License - Version 1.0 - August 17th, 2003

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import Data.List
import Data.Foldable
import Data.Traversable
import qualified Data.Bifunctor as Bifunctor

pairUp :: [a] -> ([(a,a)],Maybe a)
pairUp [] = ([],Nothing)
pairUp [a] = ([],Just a)
pairUp (a:b:t) = Bifunctor.first ((a,b) :) $ pairUp t

sortPair :: Monad m => (a -> a -> m Bool) -> (a,a) -> m (a,a)
sortPair compare (left,right) = do
 less <- compare left right
 if less then
  return (left,right)
 else 
  return (right,left)

indexes :: (Ord a, Enum a, Num a) => a -> [(a, a)]
indexes size = distribute differences 0 where
 differences = map snd $ iterate (\(i,a) -> (i+1,2^(i + 2) - a)) (0,2)
 distribute ~(box:_) current | current + box >= size = []
 distribute ~(box:rest) current = reverse [(current + box + 2, x) | x <- [current .. current + box - 1]] ++ distribute rest (current + box)
 
 
binaryInsert :: Monad m => (a -> a -> m Bool) -> Int -> Int -> a -> [a] -> m [a]
binaryInsert compare start end item list | start == end = return $ initial ++ [item] ++ final where
 (initial,final) = splitAt start list
binaryInsert compare start end item list = do
 let center = (start + end) `quot` 2
 less <- compare item (list !! center)
 if less then
  binaryInsert compare start center item list
 else
  binaryInsert compare (center + 1) end item list

-- https://en.wikipedia.org/wiki/Merge-insertion_sort
mergeInsertSort :: Monad m => (a -> a -> m Bool) -> [a] -> m [a]
mergeInsertSort _ [] = return []
mergeInsertSort _ [a] = return [a]
mergeInsertSort compare list = do
 let (paired,final) = pairUp list
 paired' <- traverse (sortPair compare) paired
 ~((first,second):paired'') <- mergeInsertSort (\(_,left) (_,right) -> compare left right) paired'
 let sorted = first : second : map snd paired''
 let unsorted = map fst paired''
 let order = indexes (length unsorted)
 let unsorted' = maybe unsorted (\final -> unsorted ++ [final]) final
 let order' = order ++ [ (length sorted + i,i) | i <- [length order .. length unsorted' - 1]]
 let queue = (map . Bifunctor.second) (unsorted' !!) order'
 foldlM (flip $ uncurry $ binaryInsert compare 0) sorted queue