```-- |
-- Module      : System.Random.Shuffle
-- Copyright   : (c) 2009 Oleg Kiselyov, Manlio Perillo
--
--
{-# OPTIONS_GHC -funbox-strict-fields #-}

module System.Random.Shuffle
(
shuffle
, shuffle'
, shuffleM
) where

import Data.Function (fix)
import System.Random (RandomGen, randomR)

-- A complete binary tree, of leaves and internal nodes.
-- Internal node: Node card l r
-- where card is the number of leaves under the node.
-- Invariant: card >=2. All internal tree nodes are always full.
data Tree a = Leaf !a
| Node !Int !(Tree a) !(Tree a)
deriving Show

-- Convert a sequence (e1...en) to a complete binary tree
buildTree :: [a] -> Tree a
buildTree = (fix growLevel) . (map Leaf)
where
growLevel _ [node] = node
growLevel self l = self \$ inner l

inner [] = []
inner [e] = [e]
inner (e1 : e2 : es) = e1 `seq` e2 `seq` (join e1 e2) : inner es

join l@(Leaf _)       r@(Leaf _)       = Node 2 l r
join l@(Node ct _ _)  r@(Leaf _)       = Node (ct + 1) l r
join l@(Leaf _)       r@(Node ct _ _)  = Node (ct + 1) l r
join l@(Node ctl _ _) r@(Node ctr _ _) = Node (ctl + ctr) l r

-- |Given a sequence (e1,...en) to shuffle, and a sequence
-- (r1,...r[n-1]) of numbers such that r[i] is an independent sample
-- from a uniform random distribution [0..n-i], compute the
-- corresponding permutation of the input sequence.
shuffle :: [a] -> [Int] -> [a]
shuffle elements = shuffleTree (buildTree elements)
where
shuffleTree (Leaf e) [] = [e]
shuffleTree tree (r : rs) =
let (b, rest) = extractTree r tree
in b : (shuffleTree rest rs)
shuffleTree _ _ = error "[shuffle] called with lists of different lengths"

-- Extracts the n-th element from the tree and returns
-- that element, paired with a tree with the element
-- deleted.
-- The function maintains the invariant of the completeness
-- of the tree: all internal nodes are always full.
extractTree 0 (Node _ (Leaf e) r) = (e, r)
extractTree 1 (Node 2 (Leaf l) (Leaf r)) = (r, Leaf l)
extractTree n (Node c (Leaf l) r) =
let (e, r') = extractTree (n - 1) r
in (e, Node (c - 1) (Leaf l) r')

extractTree n (Node n' l (Leaf e))
| n + 1 == n' = (e, l)

extractTree n (Node c l@(Node cl _ _) r)
| n < cl = let (e, l') = extractTree n l
in (e, Node (c - 1) l' r)
| otherwise = let (e, r') = extractTree (n - cl) r
in (e, Node (c - 1) l r')
extractTree _ _ = error "[extractTree] impossible"

-- |Given a sequence (e1,...en) to shuffle, its length, and a random
-- generator, compute the corresponding permutation of the input
-- sequence.
shuffle' :: RandomGen gen => [a] -> Int -> gen -> [a]
shuffle' elements len = shuffle elements . rseq len
where
-- The sequence (r1,...r[n-1]) of numbers such that r[i] is an
-- independent sample from a uniform random distribution
-- [0..n-i]
rseq :: RandomGen gen => Int -> gen -> [Int]
rseq n = fst . unzip . rseq' (n - 1)
where
rseq' :: RandomGen gen => Int -> gen -> [(Int, gen)]
rseq' 0 _ = []
rseq' i gen = (j, gen) : rseq' (i - 1) gen'
where
(j, gen') = randomR (0, i) gen

-- |shuffle' wrapped in a random monad
shuffleM :: (MonadRandom m) => [a] -> m [a]
shuffleM elements
| null elements = return []
| otherwise     = liftM (shuffle elements) (rseqM (length elements - 1))
where
rseqM :: (MonadRandom m) => Int -> m [Int]
rseqM 0 = return []
rseqM i = liftM2 (:) (getRandomR (0, i)) (rseqM (i - 1))
```