# Haskell merge sort inversions

I'm working on a task from the one of Coursera programming courses. The purpose is to estimate the amount of inversions that are required to sort the array of numbers.

Also I'm trying to learn some Haskell basics, here is my solution:

fsthalf :: [a] -> [a]
fsthalf xs = take (length xs div 2) xs

sndhalf :: [a] -> [a]
sndhalf xs = drop (length xs div 2) xs

union :: ([Int], Int) -> ([Int], Int) -> ([Int], Int)
union (a, ac) (b, bc) = (a++b, ac+bc)

merge :: ([Int], Int) -> ([Int], Int) -> ([Int], Int)
merge ([], ac) (bs, bc) = (bs, ac + bc)
merge (as, ac) ([], bc) = (as, ac + bc)
merge (a:as, ac) (b:bs, bc)
| a <= b = ([a], 0) union merge (as, ac) (b:bs, bc)
| otherwise = ([b], length(a:as)) union merge (a:as, ac) (bs, bc)

mergeSort :: ([Int], Int) -> ([Int], Int)
mergeSort (s, c)
| length s elem [0, 1]  = (s, c)
| otherwise = merge (mergeSort (fsthalf s, c)) (mergeSort (sndhalf s, c))

main :: IO()
main = do
_ <- readLn :: IO Int
rawValues <- getLine
let intValues = map read $words rawValues :: [Int] print$ snd \$ mergeSort (intValues, 0)


It seems to work properly, but exceeds time limits. It takes from 9 to 10 seconds to sort the longest possible list of integers (100000 values), while the limit is 6 seconds. Could anyone please point out some possible optimizations for this solution.

• Please don't use images of text. They're not accessible. Instead, type in the text itself. Commented Feb 25, 2018 at 3:21

## 1 Answer

General review tips:

• It'd be better to document your code. In particular functions like

union :: ([Int], Int) -> ([Int], Int) -> ([Int], Int)


need documentation - it's not clear which Int has what purpose at all without examining the source.

• Some of your functions are polymorphic (sndhalf), others not. Why? I'd suggest to have all types consistent. Either just sort Ints, or make the types as generic as possible, for example

union :: (Ord a) => ([a], Int) -> ([a], Int) -> ([a], Int)


Notice how this generality also makes the types easier to understand.

• I'd suggest to wrap list + its inversion count into a newtype such as:

newtype List a = List [a] Int


This makes the code better structured, and also allows you to use smart constructors, if you decide so.

• Investigate Haskell's sort for inspiration.

Now specific to performance: - For each pass you traverse the list several times, which can impact the performance considerably. In particular: take and length in fsthalf, the same in sndhalf. Also length in mergeSort. This makes it 5 traversals before the merging pass. I'd try to reduce this number.

• Matching on length is wasteful, if you need just to distinguish if it's 0 or 1, you traverse the whole list needlessly:

mergeSort (s, c) | length s elem [0, 1]  = (s, c)


Instead:

mergeSort ([], _) = ([], 0)
mergeSort (xs@[_], _) = (xs, 1)
...

• Make sure you're compiling with -O.

I hope this helps!