# Random shuffle

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## Revision as of 19:27, 11 January 2011

## Contents |

## 1 The problem

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

## 2 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]

## 3 Other implemenations

### 3.1 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)

### 3.2 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"]