{-| Purely functional top-down splay sets. * D.D. Sleator and R.E. Rarjan, \"Self-Adjusting Binary Search Tree\", Journal of the Association for Computing Machinery, Vol 32, No 3, July 1985, pp 652-686. <http://www.cs.cmu.edu/~sleator/papers/self-adjusting.pdf> -} module Data.Set.Splay ( -- * Data structures Splay(..) -- * Creating sets , empty , singleton , insert , fromList -- * Converting a list , toList -- * Membership , member -- * Deleting , delete , deleteMin , deleteMax -- * Checking , null -- * Set operations , union , intersection , difference -- * Helper functions , split , minimum , maximum , valid , (===) , showSet , printSet ) where import Data.List (foldl') import Prelude hiding (minimum, maximum, null) ---------------------------------------------------------------- data Splay a = Leaf | Node (Splay a) a (Splay a) deriving Show instance (Eq a) => Eq (Splay a) where t1 == t2 = toList t1 == toList t2 {-| Checking if two splay sets are exactly the same shape. -} (===) :: Eq a => Splay a -> Splay a -> Bool Leaf === Leaf = True (Node l1 x1 r1) === (Node l2 x2 r2) = x1 == x2 && l1 === l2 && r1 === r2 _ === _ = False ---------------------------------------------------------------- {-| Splitting smaller and bigger with splay. Since this is a set implementation, members must be unique. -} split :: Ord a => a -> Splay a -> (Splay a, Bool, Splay a) split _ Leaf = (Leaf,False,Leaf) split k x@(Node xl xk xr) = case compare k xk of EQ -> (xl, True, xr) GT -> case xr of Leaf -> (x, False, Leaf) Node yl yk yr -> case compare k yk of EQ -> (Node xl xk yl, True, yr) -- R :zig GT -> let (lt, b, gt) = split k yr -- RR :zig zag in (Node (Node xl xk yl) yk lt, b, gt) LT -> let (lt, b, gt) = split k yl in (Node xl xk lt, b, Node gt yk yr) -- RL :zig zig LT -> case xl of Leaf -> (Leaf, False, x) Node yl yk yr -> case compare k yk of EQ -> (yl, True, Node yr xk xr) -- L :zig GT -> let (lt, b, gt) = split k yr -- LR :zig zag in (Node yl yk lt, b, Node gt xk xr) LT -> let (lt, b, gt) = split k yl -- LL :zig zig in (lt, b, Node gt yk (Node yr xk xr)) ---------------------------------------------------------------- {-| Empty set. -} empty :: Splay a empty = Leaf {-| See if the splay set is empty. >>> Data.Set.Splay.null empty True >>> Data.Set.Splay.null (singleton 1) False -} null :: Splay a -> Bool null Leaf = True null _ = False {-| Singleton set. -} singleton :: a -> Splay a singleton x = Node Leaf x Leaf ---------------------------------------------------------------- {-| Insertion. >>> insert 5 (fromList [5,3]) == fromList [3,5] True >>> insert 7 (fromList [5,3]) == fromList [3,5,7] True >>> insert 5 empty == singleton 5 True -} insert :: Ord a => a -> Splay a -> Splay a insert x t = Node l x r where (l,_,r) = split x t ---------------------------------------------------------------- {-| Creating a set from a list. >>> empty == fromList [] True >>> singleton 'a' == fromList ['a'] True >>> fromList [5,3,5] == fromList [5,3] True -} fromList :: Ord a => [a] -> Splay a fromList = foldl' (flip insert) empty ---------------------------------------------------------------- {-| Creating a list from a set. >>> toList (fromList [5,3]) [3,5] >>> toList empty [] -} toList :: Splay a -> [a] toList t = inorder t [] where inorder Leaf xs = xs inorder (Node l x r) xs = inorder l (x : inorder r xs) ---------------------------------------------------------------- {-| Checking if this element is a member of a set? >>> fst $ member 5 (fromList [5,3]) True >>> fst $ member 1 (fromList [5,3]) False -} member :: Ord a => a -> Splay a -> (Bool, Splay a) member x t = case split x t of (l,True,r) -> (True, Node l x r) (Leaf,_,r) -> (False, r) (l,_,Leaf) -> (False, l) (l,_,r) -> let (m,l') = deleteMax l in (False, Node l' m r) ---------------------------------------------------------------- {-| Finding the minimum element. >>> fst $ minimum (fromList [3,5,1]) 1 >>> minimum empty *** Exception: minimum -} minimum :: Splay a -> (a, Splay a) minimum Leaf = error "minimum" minimum t = let (x,mt) = deleteMin t in (x, Node Leaf x mt) {-| Finding the maximum element. >>> fst $ maximum (fromList [3,5,1]) 5 >>> maximum empty *** Exception: maximum -} maximum :: Splay a -> (a, Splay a) maximum Leaf = error "maximum" maximum t = let (x,mt) = deleteMax t in (x, Node mt x Leaf) ---------------------------------------------------------------- {-| Deleting the minimum element. >>> snd (deleteMin (fromList [5,3,7])) == fromList [5,7] True >>> deleteMin empty *** Exception: deleteMin -} deleteMin :: Splay a -> (a, Splay a) deleteMin Leaf = error "deleteMin" deleteMin (Node Leaf x r) = (x,r) deleteMin (Node (Node Leaf lx lr) x r) = (lx, Node lr x r) deleteMin (Node (Node ll lx lr) x r) = let (k,mt) = deleteMin ll in (k, Node mt lx (Node lr x r)) {-| Deleting the maximum >>> snd (deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")])) == fromList [(3,"b"), (5,"a")] True >>> deleteMax empty *** Exception: deleteMax -} deleteMax :: Splay a -> (a, Splay a) deleteMax Leaf = error "deleteMax" deleteMax (Node l x Leaf) = (x,l) deleteMax (Node l x (Node rl rx Leaf)) = (rx, Node l x rl) deleteMax (Node l x (Node rl rx rr)) = let (k,mt) = deleteMax rr in (k, Node (Node l x rl) rx mt) ---------------------------------------------------------------- {-| Deleting this element from a set. >>> delete 5 (fromList [5,3]) == singleton 3 True >>> delete 7 (fromList [5,3]) == fromList [3,5] True >>> delete 5 empty == empty True -} delete :: Ord a => a -> Splay a -> Splay a delete x t = case split x t of (l, True, r) -> union l r _ -> t ---------------------------------------------------------------- {-| Creating a union set from two sets. >>> union (fromList [5,3]) (fromList [5,7]) == fromList [3,5,7] True -} union :: Ord a => Splay a -> Splay a -> Splay a union Leaf t = t union (Node a x b) t = Node (union ta a) x (union tb b) where (ta,_,tb) = split x t {-| Creating a intersection set from sets. >>> intersection (fromList [5,3]) (fromList [5,7]) == singleton 5 True -} intersection :: Ord a => Splay a -> Splay a -> Splay a intersection Leaf _ = Leaf intersection _ Leaf = Leaf intersection t1 (Node l x r) = case split x t1 of (l', True, r') -> Node (intersection l' l) x (intersection r' r) (l', False, r') -> union (intersection l' l) (intersection r' r) {-| Creating a difference set from sets. >>> difference (fromList [5,3]) (fromList [5,7]) == singleton 3 True -} difference :: Ord a => Splay a -> Splay a -> Splay a difference Leaf _ = Leaf difference t1 Leaf = t1 difference t1 (Node l x r) = union (difference l' l) (difference r' r) where (l',_,r') = split x t1 ---------------------------------------------------------------- -- Basic operations ---------------------------------------------------------------- {-| Checking validity of a set. -} valid :: Ord a => Splay a -> Bool valid t = isOrdered t isOrdered :: Ord a => Splay a -> Bool isOrdered t = ordered $ toList t where ordered [] = True ordered [_] = True ordered (x:y:xys) = x < y && ordered (y:xys) showSet :: Show a => Splay a -> String showSet = showSet' "" showSet' :: Show a => String -> Splay a -> String showSet' _ Leaf = "\n" showSet' pref (Node l x r) = show x ++ "\n" ++ pref ++ "+ " ++ showSet' pref' l ++ pref ++ "+ " ++ showSet' pref' r where pref' = " " ++ pref printSet :: Show a => Splay a -> IO () printSet = putStr . showSet {- Demo: http://www.link.cs.cmu.edu/splay/ Paper: http://www.cs.cmu.edu/~sleator/papers/self-adjusting.pdf TopDown: http://www.cs.umbc.edu/courses/undergraduate/341/fall02/Lectures/Splay/TopDownSplay.ppt Blog: http://chasen.org/~daiti-m/diary/?20061223 http://www.geocities.jp/m_hiroi/clisp/clispb07.html fromList minimum delMin member Blanced Tree N log N log N log N log N Skew Heap N log N 1 log N(???) N/A Splay Heap N log N or A(N)? log N or A(N)? log N or A(N)? -}