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+{- |
+Module      : System.Random.Shuffle
+Copyright   : (c) 2009 Oleg Kiselyov, Manlio Perillo
+License     : BSD3 (see LICENSE file)
+
+<http://okmij.org/ftp/Haskell/perfect-shuffle.txt>
+
+
+Example:
+
+    import System.Random (newStdGen)
+    import System.Random.Shuffle (shuffle')
+
+    main = do
+      rng <- newStdGen
+      let xs = [1,2,3,4,5]
+      print $ shuffle' xs (length xs) rng
+-}
+{-# OPTIONS_GHC -funbox-strict-fields #-}
+
+module System.Random.Shuffle
+  ( shuffle
+  , shuffle'
+  , shuffleM
+  )
+where
+
+import           Data.Function                  ( fix )
+import           System.Random                  ( RandomGen
+                                                , randomR
+                                                )
+import           Control.Monad                  ( liftM
+                                                , liftM2
+                                                )
+import           Control.Monad.Random           ( MonadRandom
+                                                , getRandomR
+                                                )
+
+
+-- | 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))