# 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.

## Packages

There are ready made packages available from Hackage:

## Imperative algorithm

The standard imperative algorithm can be implemented as follows:

```import System.Random
import Data.Array.IO

-- | 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 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 System.Random
import Data.Array.ST
import GHC.Arr

shuffle :: RandomGen g => [a] -> Rand g [a]
shuffle xs = do
let l = length xs
rands <- forM [0..(l-2)] \$ \i -> getRandomR (i, l-1)
let ar = runSTArray \$ do
ar <- thawSTArray \$ listArray (0, l-1) xs
forM_ (zip [0..] 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 => (Map Int a, g) -> (Int, a) -> (Map Int a, g)
fisherYatesStep (m, gen) (i, x) = ((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 =
toElems \$ foldl fisherYatesStep (initial (head l) gen) (numerate (tail l))
where
toElems (x, y) = (elems x, y)
numerate = zip [1..]
initial x gen = (singleton 0 x, gen)
```

### Drawing without replacement

• 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)