Random shuffle

The problem

Shuffling a list, i.e. creating a random permutation, is not easy to do correctly. Each permutation should have the same probability.

Imperative algorithm

The standard imperative algorithm can be implemented as follows:

import System.Random
import Data.Array.IO
import Control.Monad

-- | Randomly shuffle a list
--   /O(N)/
shuffle :: [a] -> IO [a]
shuffle xs = do
        ar <- newArray n xs
        forM [1..n] $ \i -> do
            j <- randomRIO (i,n)
            vi <- readArray ar i
            vj <- readArray ar j
            writeArray ar j vi
            return vj
  where
    n = length xs
    newArray :: Int -> [a] -> IO (IOArray Int a)
    newArray n xs =  newListArray (1,n) xs

Or one can use ST to avoid needing IO:

import System.Random
import Data.Array.ST
import Control.Monad
import Control.Monad.ST
import Data.STRef

-- | Randomly shuffle a list without the IO Monad
--   /O(N)/
shuffle' :: [a] -> StdGen -> ([a],StdGen)
shuffle' xs gen = runST (do
        g <- newSTRef gen
        let randomRST lohi = do
              (a,s') <- liftM (randomR lohi) (readSTRef g)
              writeSTRef g s'
              return a
        ar <- newArray n xs
        xs' <- forM [1..n] $ \i -> do
                j <- randomRST (i,n)
                vi <- readArray ar i
                vj <- readArray ar j
                writeArray ar j vi
                return vj
        gen' <- readSTRef g
        return (xs',gen'))
  where
    n = length xs
    newArray :: Int -> [a] -> ST s (STArray s Int a)
    newArray n xs =  newListArray (1,n) xs

And if you are using IO's hidden StdGen you can wrap this as usual:

shuffleIO :: [a] -> IO [a]
shuffleIO xs = getStdRandom (shuffle' xs)

This is a lot simpler than the purely functional algorithm linked below.

Here's a variation using the MonadRandom package:

import Control.Monad
import Control.Monad.ST
import Control.Monad.Random
import System.Random
import Data.Array.ST
import GHC.Arr

shuffle :: RandomGen g => [a] -> Rand g [a]
shuffle xs = do
    let l = length xs
    rands <- take l `fmap` getRandomRs (0, l-1)
    let ar = runSTArray $ do
        ar <- thawSTArray $ listArray (0, l-1) xs
        forM_ (zip [0..(l-1)] rands) $ \(i, j) -> do
            vi <- readSTArray ar i
            vj <- readSTArray ar j
            writeSTArray ar j vi
            writeSTArray ar i vj
        return ar
    return (elems ar)

*Main> evalRandIO (shuffle [1..10])
[6,5,1,7,10,4,9,2,8,3]

Other implemenations

Purely functional

• Using Data.Map, O(n * log n)

import System.Random
import Data.Map

fisherYatesStep :: RandomGen g => (Int, Map Int a, g) -> a -> (Int, Map Int a, g)
fisherYatesStep (i, m, gen) x = (i+1, (insert j x . insert i (m ! j)) m, gen')
  where
    (j, gen') = randomR (0, i) gen

fisherYates :: RandomGen g => g -> [a] -> ([a], g)
fisherYates gen [] = ([], gen)
fisherYates gen l = (elems m, gen')
  where
    (_, m, gen') = foldl fisherYatesStep (1, singleton 0 (head l), gen) (tail l)

• Purely functional O(n log n) random shuffle algorithm.

Drawing without replacement

• uses New_monads/MonadRandom
• allows you to not shuffle the entire list but only part of it (drawing elements without replacement)
• allows you to take multiple drawings/shufflings at once, which can save some array building

{- | @grabble xs m n@ is /O(m*n')/, where @n' = min n (length xs)@
     Chooses @n@ elements from @xs@, without putting back,
     and that @m@ times. -}
grabble :: MonadRandom m => [a] -> Int -> Int -> m [[a]]
grabble xs m n = do
    swapss <- replicateM m $ forM [0 .. min (maxIx - 1) n] $ \i -> do
                j <- getRandomR (i, maxIx)
                return (i, j)
    return $ map (take n . swapElems xs) swapss
    where
        maxIx   = length xs - 1

grabbleOnce :: MonadRandom m => [a] -> Int -> m [a]
grabbleOnce xs n = head `liftM` grabble xs 1 n

swapElems  :: [a] -> [(Int, Int)] -> [a]
swapElems xs swaps = elems $ runSTArray (do
    arr <- newListArray (0, maxIx) xs
    mapM_ (swap arr) swaps
    return arr)
    where
        maxIx   = length xs - 1
        swap arr (i,j) = do
            vi <- readArray arr i
            vj <- readArray arr j
            writeArray arr i vj
            writeArray arr j vi

So e.g.

*Main MonadRandom Random> evalRand (grabble "abcdef" 6 3) (mkStdGen 0)
["fbd","efb","bef","adc","cef","eac"]
*Main MonadRandom Random> grabble "abcdef" 6 3
["fce","dfa","ebf","edb","cea","dbc"]
*Main MonadRandom Random> grabble "abcdef" 6 3
["cbf","dec","edb","fae","bda","cde"]