{-# LANGUAGE CPP #-} #if __GLASGOW_HASKELL__ {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-} #endif #if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703 {-# LANGUAGE Trustworthy #-} #endif ----------------------------------------------------------------------------- -- | -- Module : Data.Set.Base -- Copyright : (c) Daan Leijen 2002 -- License : BSD-style -- Maintainer : libraries@haskell.org -- Stability : provisional -- Portability : portable -- -- An efficient implementation of sets. -- -- These modules are intended to be imported qualified, to avoid name -- clashes with Prelude functions, e.g. -- -- > import Data.Set (Set) -- > import qualified Data.Set as Set -- -- The implementation of 'Set' is based on /size balanced/ binary trees (or -- trees of /bounded balance/) as described by: -- -- * Stephen Adams, \"/Efficient sets: a balancing act/\", -- Journal of Functional Programming 3(4):553-562, October 1993, -- <http://www.swiss.ai.mit.edu/~adams/BB/>. -- -- * J. Nievergelt and E.M. Reingold, -- \"/Binary search trees of bounded balance/\", -- SIAM journal of computing 2(1), March 1973. -- -- Note that the implementation is /left-biased/ -- the elements of a -- first argument are always preferred to the second, for example in -- 'union' or 'insert'. Of course, left-biasing can only be observed -- when equality is an equivalence relation instead of structural -- equality. ----------------------------------------------------------------------------- -- [Note: Using INLINABLE] -- ~~~~~~~~~~~~~~~~~~~~~~~ -- It is crucial to the performance that the functions specialize on the Ord -- type when possible. GHC 7.0 and higher does this by itself when it sees th -- unfolding of a function -- that is why all public functions are marked -- INLINABLE (that exposes the unfolding). -- [Note: Using INLINE] -- ~~~~~~~~~~~~~~~~~~~~ -- For other compilers and GHC pre 7.0, we mark some of the functions INLINE. -- We mark the functions that just navigate down the tree (lookup, insert, -- delete and similar). That navigation code gets inlined and thus specialized -- when possible. There is a price to pay -- code growth. The code INLINED is -- therefore only the tree navigation, all the real work (rebalancing) is not -- INLINED by using a NOINLINE. -- -- All methods marked INLINE have to be nonrecursive -- a 'go' function doing -- the real work is provided. -- [Note: Type of local 'go' function] -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- If the local 'go' function uses an Ord class, it sometimes heap-allocates -- the Ord dictionary when the 'go' function does not have explicit type. -- In that case we give 'go' explicit type. But this slightly decrease -- performance, as the resulting 'go' function can float out to top level. -- [Note: Local 'go' functions and capturing] -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- As opposed to IntSet, when 'go' function captures an argument, increased -- heap-allocation can occur: sometimes in a polymorphic function, the 'go' -- floats out of its enclosing function and then it heap-allocates the -- dictionary and the argument. Maybe it floats out too late and strictness -- analyzer cannot see that these could be passed on stack. -- -- For example, change 'member' so that its local 'go' function is not passing -- argument x and then look at the resulting code for hedgeInt. -- [Note: Order of constructors] -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- The order of constructors of Set matters when considering performance. -- Currently in GHC 7.0, when type has 2 constructors, a forward conditional -- jump is made when successfully matching second constructor. Successful match -- of first constructor results in the forward jump not taken. -- On GHC 7.0, reordering constructors from Tip | Bin to Bin | Tip -- improves the benchmark by up to 10% on x86. module Data.Set.Base ( -- * Set type Set(..) -- instance Eq,Ord,Show,Read,Data,Typeable -- * Operators , (\\) -- * Query , null , size , member , notMember , lookupLT , lookupGT , lookupLE , lookupGE , isSubsetOf , isProperSubsetOf -- * Construction , empty , singleton , insert , delete -- * Combine , union , unions , difference , intersection -- * Filter , filter , partition , split , splitMember -- * Map , map , mapMonotonic -- * Folds , foldr , foldl -- ** Strict folds , foldr' , foldl' -- ** Legacy folds , fold -- * Min\/Max , findMin , findMax , deleteMin , deleteMax , deleteFindMin , deleteFindMax , maxView , minView -- * Conversion -- ** List , elems , toList , fromList -- ** Ordered list , toAscList , toDescList , fromAscList , fromDistinctAscList -- * Debugging , showTree , showTreeWith , valid -- Internals (for testing) , bin , balanced , join , merge ) where import Prelude hiding (filter,foldl,foldr,null,map) import qualified Data.List as List import Data.Monoid (Monoid(..)) import qualified Data.Foldable as Foldable import Data.Typeable import Control.DeepSeq (NFData(rnf)) #if __GLASGOW_HASKELL__ import GHC.Exts ( build ) import Text.Read import Data.Data #endif -- Use macros to define strictness of functions. -- STRICT_x_OF_y denotes an y-ary function strict in the x-th parameter. -- We do not use BangPatterns, because they are not in any standard and we -- want the compilers to be compiled by as many compilers as possible. #define STRICT_1_OF_2(fn) fn arg _ | arg `seq` False = undefined #define STRICT_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined {-------------------------------------------------------------------- Operators --------------------------------------------------------------------} infixl 9 \\ -- -- | /O(n+m)/. See 'difference'. (\\) :: Ord a => Set a -> Set a -> Set a m1 \\ m2 = difference m1 m2 #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE (\\) #-} #endif {-------------------------------------------------------------------- Sets are size balanced trees --------------------------------------------------------------------} -- | A set of values @a@. -- See Note: Order of constructors data Set a = Bin {-# UNPACK #-} !Size !a !(Set a) !(Set a) | Tip type Size = Int instance Ord a => Monoid (Set a) where mempty = empty mappend = union mconcat = unions instance Foldable.Foldable Set where fold Tip = mempty fold (Bin _ k l r) = Foldable.fold l `mappend` k `mappend` Foldable.fold r foldr = foldr foldl = foldl foldMap _ Tip = mempty foldMap f (Bin _ k l r) = Foldable.foldMap f l `mappend` f k `mappend` Foldable.foldMap f r #if __GLASGOW_HASKELL__ {-------------------------------------------------------------------- A Data instance --------------------------------------------------------------------} -- This instance preserves data abstraction at the cost of inefficiency. -- We omit reflection services for the sake of data abstraction. instance (Data a, Ord a) => Data (Set a) where gfoldl f z set = z fromList `f` (toList set) toConstr _ = error "toConstr" gunfold _ _ = error "gunfold" dataTypeOf _ = mkNoRepType "Data.Set.Set" dataCast1 f = gcast1 f #endif {-------------------------------------------------------------------- Query --------------------------------------------------------------------} -- | /O(1)/. Is this the empty set? null :: Set a -> Bool null Tip = True null (Bin {}) = False {-# INLINE null #-} -- | /O(1)/. The number of elements in the set. size :: Set a -> Int size Tip = 0 size (Bin sz _ _ _) = sz {-# INLINE size #-} -- | /O(log n)/. Is the element in the set? member :: Ord a => a -> Set a -> Bool member = go where STRICT_1_OF_2(go) go _ Tip = False go x (Bin _ y l r) = case compare x y of LT -> go x l GT -> go x r EQ -> True #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE member #-} #else {-# INLINE member #-} #endif -- | /O(log n)/. Is the element not in the set? notMember :: Ord a => a -> Set a -> Bool notMember a t = not $ member a t #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE notMember #-} #else {-# INLINE notMember #-} #endif -- | /O(log n)/. Find largest element smaller than the given one. -- -- > lookupLT 3 (fromList [3, 5]) == Nothing -- > lookupLT 5 (fromList [3, 5]) == Just 3 lookupLT :: Ord a => a -> Set a -> Maybe a lookupLT = goNothing where STRICT_1_OF_2(goNothing) goNothing _ Tip = Nothing goNothing x (Bin _ y l r) | x <= y = goNothing x l | otherwise = goJust x y r STRICT_1_OF_3(goJust) goJust _ best Tip = Just best goJust x best (Bin _ y l r) | x <= y = goJust x best l | otherwise = goJust x y r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE lookupLT #-} #else {-# INLINE lookupLT #-} #endif -- | /O(log n)/. Find smallest element greater than the given one. -- -- > lookupGT 4 (fromList [3, 5]) == Just 5 -- > lookupGT 5 (fromList [3, 5]) == Nothing lookupGT :: Ord a => a -> Set a -> Maybe a lookupGT = goNothing where STRICT_1_OF_2(goNothing) goNothing _ Tip = Nothing goNothing x (Bin _ y l r) | x < y = goJust x y l | otherwise = goNothing x r STRICT_1_OF_3(goJust) goJust _ best Tip = Just best goJust x best (Bin _ y l r) | x < y = goJust x y l | otherwise = goJust x best r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE lookupGT #-} #else {-# INLINE lookupGT #-} #endif -- | /O(log n)/. Find largest element smaller or equal to the given one. -- -- > lookupLE 2 (fromList [3, 5]) == Nothing -- > lookupLE 4 (fromList [3, 5]) == Just 3 -- > lookupLE 5 (fromList [3, 5]) == Just 5 lookupLE :: Ord a => a -> Set a -> Maybe a lookupLE = goNothing where STRICT_1_OF_2(goNothing) goNothing _ Tip = Nothing goNothing x (Bin _ y l r) = case compare x y of LT -> goNothing x l EQ -> Just y GT -> goJust x y r STRICT_1_OF_3(goJust) goJust _ best Tip = Just best goJust x best (Bin _ y l r) = case compare x y of LT -> goJust x best l EQ -> Just y GT -> goJust x y r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE lookupLE #-} #else {-# INLINE lookupLE #-} #endif -- | /O(log n)/. Find smallest element greater or equal to the given one. -- -- > lookupGE 3 (fromList [3, 5]) == Just 3 -- > lookupGE 4 (fromList [3, 5]) == Just 5 -- > lookupGE 6 (fromList [3, 5]) == Nothing lookupGE :: Ord a => a -> Set a -> Maybe a lookupGE = goNothing where STRICT_1_OF_2(goNothing) goNothing _ Tip = Nothing goNothing x (Bin _ y l r) = case compare x y of LT -> goJust x y l EQ -> Just y GT -> goNothing x r STRICT_1_OF_3(goJust) goJust _ best Tip = Just best goJust x best (Bin _ y l r) = case compare x y of LT -> goJust x y l EQ -> Just y GT -> goJust x best r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE lookupGE #-} #else {-# INLINE lookupGE #-} #endif {-------------------------------------------------------------------- Construction --------------------------------------------------------------------} -- | /O(1)/. The empty set. empty :: Set a empty = Tip {-# INLINE empty #-} -- | /O(1)/. Create a singleton set. singleton :: a -> Set a singleton x = Bin 1 x Tip Tip {-# INLINE singleton #-} {-------------------------------------------------------------------- Insertion, Deletion --------------------------------------------------------------------} -- | /O(log n)/. Insert an element in a set. -- If the set already contains an element equal to the given value, -- it is replaced with the new value. -- See Note: Type of local 'go' function insert :: Ord a => a -> Set a -> Set a insert = go where go :: Ord a => a -> Set a -> Set a STRICT_1_OF_2(go) go x Tip = singleton x go x (Bin sz y l r) = case compare x y of LT -> balanceL y (go x l) r GT -> balanceR y l (go x r) EQ -> Bin sz x l r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE insert #-} #else {-# INLINE insert #-} #endif -- Insert an element to the set only if it is not in the set. -- Used by `union`. -- See Note: Type of local 'go' function insertR :: Ord a => a -> Set a -> Set a insertR = go where go :: Ord a => a -> Set a -> Set a STRICT_1_OF_2(go) go x Tip = singleton x go x t@(Bin _ y l r) = case compare x y of LT -> balanceL y (go x l) r GT -> balanceR y l (go x r) EQ -> t #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE insertR #-} #else {-# INLINE insertR #-} #endif -- | /O(log n)/. Delete an element from a set. -- See Note: Type of local 'go' function delete :: Ord a => a -> Set a -> Set a delete = go where go :: Ord a => a -> Set a -> Set a STRICT_1_OF_2(go) go _ Tip = Tip go x (Bin _ y l r) = case compare x y of LT -> balanceR y (go x l) r GT -> balanceL y l (go x r) EQ -> glue l r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE delete #-} #else {-# INLINE delete #-} #endif {-------------------------------------------------------------------- Subset --------------------------------------------------------------------} -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal). isProperSubsetOf :: Ord a => Set a -> Set a -> Bool isProperSubsetOf s1 s2 = (size s1 < size s2) && (isSubsetOf s1 s2) #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE isProperSubsetOf #-} #endif -- | /O(n+m)/. Is this a subset? -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@. isSubsetOf :: Ord a => Set a -> Set a -> Bool isSubsetOf t1 t2 = (size t1 <= size t2) && (isSubsetOfX t1 t2) #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE isSubsetOf #-} #endif isSubsetOfX :: Ord a => Set a -> Set a -> Bool isSubsetOfX Tip _ = True isSubsetOfX _ Tip = False isSubsetOfX (Bin _ x l r) t = found && isSubsetOfX l lt && isSubsetOfX r gt where (lt,found,gt) = splitMember x t #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE isSubsetOfX #-} #endif {-------------------------------------------------------------------- Minimal, Maximal --------------------------------------------------------------------} -- | /O(log n)/. The minimal element of a set. findMin :: Set a -> a findMin (Bin _ x Tip _) = x findMin (Bin _ _ l _) = findMin l findMin Tip = error "Set.findMin: empty set has no minimal element" -- | /O(log n)/. The maximal element of a set. findMax :: Set a -> a findMax (Bin _ x _ Tip) = x findMax (Bin _ _ _ r) = findMax r findMax Tip = error "Set.findMax: empty set has no maximal element" -- | /O(log n)/. Delete the minimal element. deleteMin :: Set a -> Set a deleteMin (Bin _ _ Tip r) = r deleteMin (Bin _ x l r) = balanceR x (deleteMin l) r deleteMin Tip = Tip -- | /O(log n)/. Delete the maximal element. deleteMax :: Set a -> Set a deleteMax (Bin _ _ l Tip) = l deleteMax (Bin _ x l r) = balanceL x l (deleteMax r) deleteMax Tip = Tip {-------------------------------------------------------------------- Union. --------------------------------------------------------------------} -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@). unions :: Ord a => [Set a] -> Set a unions = foldlStrict union empty #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE unions #-} #endif -- | /O(n+m)/. The union of two sets, preferring the first set when -- equal elements are encountered. -- The implementation uses the efficient /hedge-union/ algorithm. -- Hedge-union is more efficient on (bigset `union` smallset). union :: Ord a => Set a -> Set a -> Set a union Tip t2 = t2 union t1 Tip = t1 union t1 t2 = hedgeUnion NothingS NothingS t1 t2 #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE union #-} #endif hedgeUnion :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a hedgeUnion _ _ t1 Tip = t1 hedgeUnion blo bhi Tip (Bin _ x l r) = join x (filterGt blo l) (filterLt bhi r) hedgeUnion _ _ t1 (Bin _ x Tip Tip) = insertR x t1 -- According to benchmarks, this special case increases -- performance up to 30%. It does not help in difference or intersection. hedgeUnion blo bhi (Bin _ x l r) t2 = join x (hedgeUnion blo bmi l (trim blo bmi t2)) (hedgeUnion bmi bhi r (trim bmi bhi t2)) where bmi = JustS x #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE hedgeUnion #-} #endif {-------------------------------------------------------------------- Difference --------------------------------------------------------------------} -- | /O(n+m)/. Difference of two sets. -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/. difference :: Ord a => Set a -> Set a -> Set a difference Tip _ = Tip difference t1 Tip = t1 difference t1 t2 = hedgeDiff NothingS NothingS t1 t2 #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE difference #-} #endif hedgeDiff :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a hedgeDiff _ _ Tip _ = Tip hedgeDiff blo bhi (Bin _ x l r) Tip = join x (filterGt blo l) (filterLt bhi r) hedgeDiff blo bhi t (Bin _ x l r) = merge (hedgeDiff blo bmi (trim blo bmi t) l) (hedgeDiff bmi bhi (trim bmi bhi t) r) where bmi = JustS x #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE hedgeDiff #-} #endif {-------------------------------------------------------------------- Intersection --------------------------------------------------------------------} -- | /O(n+m)/. The intersection of two sets. -- Elements of the result come from the first set, so for example -- -- > import qualified Data.Set as S -- > data AB = A | B deriving Show -- > instance Ord AB where compare _ _ = EQ -- > instance Eq AB where _ == _ = True -- > main = print (S.singleton A `S.intersection` S.singleton B, -- > S.singleton B `S.intersection` S.singleton A) -- -- prints @(fromList [A],fromList [B])@. intersection :: Ord a => Set a -> Set a -> Set a intersection Tip _ = Tip intersection _ Tip = Tip intersection t1 t2 = hedgeInt NothingS NothingS t1 t2 #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE intersection #-} #endif hedgeInt :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a -> Set a hedgeInt _ _ _ Tip = Tip hedgeInt _ _ Tip _ = Tip hedgeInt blo bhi (Bin _ x l r) t2 = let l' = hedgeInt blo bmi l (trim blo bmi t2) r' = hedgeInt bmi bhi r (trim bmi bhi t2) in if x `member` t2 then join x l' r' else merge l' r' where bmi = JustS x #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE hedgeInt #-} #endif {-------------------------------------------------------------------- Filter and partition --------------------------------------------------------------------} -- | /O(n)/. Filter all elements that satisfy the predicate. filter :: (a -> Bool) -> Set a -> Set a filter _ Tip = Tip filter p (Bin _ x l r) | p x = join x (filter p l) (filter p r) | otherwise = merge (filter p l) (filter p r) -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy -- the predicate and one with all elements that don't satisfy the predicate. -- See also 'split'. partition :: (a -> Bool) -> Set a -> (Set a,Set a) partition _ Tip = (Tip, Tip) partition p (Bin _ x l r) = case (partition p l, partition p r) of ((l1, l2), (r1, r2)) | p x -> (join x l1 r1, merge l2 r2) | otherwise -> (merge l1 r1, join x l2 r2) {---------------------------------------------------------------------- Map ----------------------------------------------------------------------} -- | /O(n*log n)/. -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@. -- -- It's worth noting that the size of the result may be smaller if, -- for some @(x,y)@, @x \/= y && f x == f y@ map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b map f = fromList . List.map f . toList #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE map #-} #endif -- | /O(n)/. The -- -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic. -- /The precondition is not checked./ -- Semi-formally, we have: -- -- > and [x < y ==> f x < f y | x <- ls, y <- ls] -- > ==> mapMonotonic f s == map f s -- > where ls = toList s mapMonotonic :: (a->b) -> Set a -> Set b mapMonotonic _ Tip = Tip mapMonotonic f (Bin sz x l r) = Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r) {-------------------------------------------------------------------- Fold --------------------------------------------------------------------} -- | /O(n)/. Fold the elements in the set using the given right-associative -- binary operator. This function is an equivalent of 'foldr' and is present -- for compatibility only. -- -- /Please note that fold will be deprecated in the future and removed./ fold :: (a -> b -> b) -> b -> Set a -> b fold = foldr {-# INLINE fold #-} -- | /O(n)/. Fold the elements in the set using the given right-associative -- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'toAscList'@. -- -- For example, -- -- > toAscList set = foldr (:) [] set foldr :: (a -> b -> b) -> b -> Set a -> b foldr f z = go z where go z' Tip = z' go z' (Bin _ x l r) = go (f x (go z' r)) l {-# INLINE foldr #-} -- | /O(n)/. A strict version of 'foldr'. Each application of the operator is -- evaluated before using the result in the next application. This -- function is strict in the starting value. foldr' :: (a -> b -> b) -> b -> Set a -> b foldr' f z = go z where STRICT_1_OF_2(go) go z' Tip = z' go z' (Bin _ x l r) = go (f x (go z' r)) l {-# INLINE foldr' #-} -- | /O(n)/. Fold the elements in the set using the given left-associative -- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'toAscList'@. -- -- For example, -- -- > toDescList set = foldl (flip (:)) [] set foldl :: (a -> b -> a) -> a -> Set b -> a foldl f z = go z where go z' Tip = z' go z' (Bin _ x l r) = go (f (go z' l) x) r {-# INLINE foldl #-} -- | /O(n)/. A strict version of 'foldl'. Each application of the operator is -- evaluated before using the result in the next application. This -- function is strict in the starting value. foldl' :: (a -> b -> a) -> a -> Set b -> a foldl' f z = go z where STRICT_1_OF_2(go) go z' Tip = z' go z' (Bin _ x l r) = go (f (go z' l) x) r {-# INLINE foldl' #-} {-------------------------------------------------------------------- List variations --------------------------------------------------------------------} -- | /O(n)/. An alias of 'toAscList'. The elements of a set in ascending order. -- Subject to list fusion. elems :: Set a -> [a] elems = toAscList {-------------------------------------------------------------------- Lists --------------------------------------------------------------------} -- | /O(n)/. Convert the set to a list of elements. Subject to list fusion. toList :: Set a -> [a] toList = toAscList -- | /O(n)/. Convert the set to an ascending list of elements. Subject to list fusion. toAscList :: Set a -> [a] toAscList = foldr (:) [] -- | /O(n)/. Convert the set to a descending list of elements. Subject to list -- fusion. toDescList :: Set a -> [a] toDescList = foldl (flip (:)) [] -- List fusion for the list generating functions. #if __GLASGOW_HASKELL__ -- The foldrFB and foldlFB are foldr and foldl equivalents, used for list fusion. -- They are important to convert unfused to{Asc,Desc}List back, see mapFB in prelude. foldrFB :: (a -> b -> b) -> b -> Set a -> b foldrFB = foldr {-# INLINE[0] foldrFB #-} foldlFB :: (a -> b -> a) -> a -> Set b -> a foldlFB = foldl {-# INLINE[0] foldlFB #-} -- Inline elems and toList, so that we need to fuse only toAscList. {-# INLINE elems #-} {-# INLINE toList #-} -- The fusion is enabled up to phase 2 included. If it does not succeed, -- convert in phase 1 the expanded to{Asc,Desc}List calls back to -- to{Asc,Desc}List. In phase 0, we inline fold{lr}FB (which were used in -- a list fusion, otherwise it would go away in phase 1), and let compiler do -- whatever it wants with to{Asc,Desc}List -- it was forbidden to inline it -- before phase 0, otherwise the fusion rules would not fire at all. {-# NOINLINE[0] toAscList #-} {-# NOINLINE[0] toDescList #-} {-# RULES "Set.toAscList" [~1] forall s . toAscList s = build (\c n -> foldrFB c n s) #-} {-# RULES "Set.toAscListBack" [1] foldrFB (:) [] = toAscList #-} {-# RULES "Set.toDescList" [~1] forall s . toDescList s = build (\c n -> foldlFB (\xs x -> c x xs) n s) #-} {-# RULES "Set.toDescListBack" [1] foldlFB (\xs x -> x : xs) [] = toDescList #-} #endif -- | /O(n*log n)/. Create a set from a list of elements. fromList :: Ord a => [a] -> Set a fromList = foldlStrict ins empty where ins t x = insert x t #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE fromList #-} #endif {-------------------------------------------------------------------- Building trees from ascending/descending lists can be done in linear time. Note that if [xs] is ascending that: fromAscList xs == fromList xs --------------------------------------------------------------------} -- | /O(n)/. Build a set from an ascending list in linear time. -- /The precondition (input list is ascending) is not checked./ fromAscList :: Eq a => [a] -> Set a fromAscList xs = fromDistinctAscList (combineEq xs) where -- [combineEq xs] combines equal elements with [const] in an ordered list [xs] combineEq xs' = case xs' of [] -> [] [x] -> [x] (x:xx) -> combineEq' x xx combineEq' z [] = [z] combineEq' z (x:xs') | z==x = combineEq' z xs' | otherwise = z:combineEq' x xs' #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE fromAscList #-} #endif -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time. -- /The precondition (input list is strictly ascending) is not checked./ fromDistinctAscList :: [a] -> Set a fromDistinctAscList xs = create const (length xs) xs where -- 1) use continutations so that we use heap space instead of stack space. -- 2) special case for n==5 to create bushier trees. create c 0 xs' = c Tip xs' create c 5 xs' = case xs' of (x1:x2:x3:x4:x5:xx) -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx _ -> error "fromDistinctAscList create 5" create c n xs' = seq nr $ create (createR nr c) nl xs' where nl = n `div` 2 nr = n - nl - 1 createR n c l (x:ys) = create (createB l x c) n ys createR _ _ _ [] = error "fromDistinctAscList createR []" createB l x c r zs = c (bin x l r) zs {-------------------------------------------------------------------- Eq converts the set to a list. In a lazy setting, this actually seems one of the faster methods to compare two trees and it is certainly the simplest :-) --------------------------------------------------------------------} instance Eq a => Eq (Set a) where t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2) {-------------------------------------------------------------------- Ord --------------------------------------------------------------------} instance Ord a => Ord (Set a) where compare s1 s2 = compare (toAscList s1) (toAscList s2) {-------------------------------------------------------------------- Show --------------------------------------------------------------------} instance Show a => Show (Set a) where showsPrec p xs = showParen (p > 10) $ showString "fromList " . shows (toList xs) {-------------------------------------------------------------------- Read --------------------------------------------------------------------} instance (Read a, Ord a) => Read (Set a) where #ifdef __GLASGOW_HASKELL__ readPrec = parens $ prec 10 $ do Ident "fromList" <- lexP xs <- readPrec return (fromList xs) readListPrec = readListPrecDefault #else readsPrec p = readParen (p > 10) $ \ r -> do ("fromList",s) <- lex r (xs,t) <- reads s return (fromList xs,t) #endif {-------------------------------------------------------------------- Typeable/Data --------------------------------------------------------------------} #include "Typeable.h" INSTANCE_TYPEABLE1(Set,setTc,"Set") {-------------------------------------------------------------------- NFData --------------------------------------------------------------------} instance NFData a => NFData (Set a) where rnf Tip = () rnf (Bin _ y l r) = rnf y `seq` rnf l `seq` rnf r {-------------------------------------------------------------------- Utility functions that return sub-ranges of the original tree. Some functions take a `Maybe value` as an argument to allow comparisons against infinite values. These are called `blow` (Nothing is -\infty) and `bhigh` (here Nothing is +\infty). We use MaybeS value, which is a Maybe strict in the Just case. [trim blow bhigh t] A tree that is either empty or where [x > blow] and [x < bhigh] for the value [x] of the root. [filterGt blow t] A tree where for all values [k]. [k > blow] [filterLt bhigh t] A tree where for all values [k]. [k < bhigh] [split k t] Returns two trees [l] and [r] where all values in [l] are <[k] and all keys in [r] are >[k]. [splitMember k t] Just like [split] but also returns whether [k] was found in the tree. --------------------------------------------------------------------} data MaybeS a = NothingS | JustS !a {-------------------------------------------------------------------- [trim blo bhi t] trims away all subtrees that surely contain no values between the range [blo] to [bhi]. The returned tree is either empty or the key of the root is between @blo@ and @bhi@. --------------------------------------------------------------------} trim :: Ord a => MaybeS a -> MaybeS a -> Set a -> Set a trim NothingS NothingS t = t trim (JustS lx) NothingS t = greater lx t where greater lo (Bin _ x _ r) | x <= lo = greater lo r greater _ t' = t' trim NothingS (JustS hx) t = lesser hx t where lesser hi (Bin _ x l _) | x >= hi = lesser hi l lesser _ t' = t' trim (JustS lx) (JustS hx) t = middle lx hx t where middle lo hi (Bin _ x _ r) | x <= lo = middle lo hi r middle lo hi (Bin _ x l _) | x >= hi = middle lo hi l middle _ _ t' = t' #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE trim #-} #endif {-------------------------------------------------------------------- [filterGt b t] filter all values >[b] from tree [t] [filterLt b t] filter all values <[b] from tree [t] --------------------------------------------------------------------} filterGt :: Ord a => MaybeS a -> Set a -> Set a filterGt NothingS t = t filterGt (JustS b) t = filter' b t where filter' _ Tip = Tip filter' b' (Bin _ x l r) = case compare b' x of LT -> join x (filter' b' l) r EQ -> r GT -> filter' b' r #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE filterGt #-} #endif filterLt :: Ord a => MaybeS a -> Set a -> Set a filterLt NothingS t = t filterLt (JustS b) t = filter' b t where filter' _ Tip = Tip filter' b' (Bin _ x l r) = case compare x b' of LT -> join x l (filter' b' r) EQ -> l GT -> filter' b' l #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE filterLt #-} #endif {-------------------------------------------------------------------- Split --------------------------------------------------------------------} -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@ -- where @set1@ comprises the elements of @set@ less than @x@ and @set2@ -- comprises the elements of @set@ greater than @x@. split :: Ord a => a -> Set a -> (Set a,Set a) split _ Tip = (Tip,Tip) split x (Bin _ y l r) = case compare x y of LT -> let (lt,gt) = split x l in (lt,join y gt r) GT -> let (lt,gt) = split x r in (join y l lt,gt) EQ -> (l,r) #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE split #-} #endif -- | /O(log n)/. Performs a 'split' but also returns whether the pivot -- element was found in the original set. splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a) splitMember _ Tip = (Tip, False, Tip) splitMember x (Bin _ y l r) = case compare x y of LT -> let (lt, found, gt) = splitMember x l in (lt, found, join y gt r) GT -> let (lt, found, gt) = splitMember x r in (join y l lt, found, gt) EQ -> (l, True, r) #if __GLASGOW_HASKELL__ >= 700 {-# INLINABLE splitMember #-} #endif {-------------------------------------------------------------------- Utility functions that maintain the balance properties of the tree. All constructors assume that all values in [l] < [x] and all values in [r] > [x], and that [l] and [r] are valid trees. In order of sophistication: [Bin sz x l r] The type constructor. [bin x l r] Maintains the correct size, assumes that both [l] and [r] are balanced with respect to each other. [balance x l r] Restores the balance and size. Assumes that the original tree was balanced and that [l] or [r] has changed by at most one element. [join x l r] Restores balance and size. Furthermore, we can construct a new tree from two trees. Both operations assume that all values in [l] < all values in [r] and that [l] and [r] are valid: [glue l r] Glues [l] and [r] together. Assumes that [l] and [r] are already balanced with respect to each other. [merge l r] Merges two trees and restores balance. Note: in contrast to Adam's paper, we use (<=) comparisons instead of (<) comparisons in [join], [merge] and [balance]. Quickcheck (on [difference]) showed that this was necessary in order to maintain the invariants. It is quite unsatisfactory that I haven't been able to find out why this is actually the case! Fortunately, it doesn't hurt to be a bit more conservative. --------------------------------------------------------------------} {-------------------------------------------------------------------- Join --------------------------------------------------------------------} join :: a -> Set a -> Set a -> Set a join x Tip r = insertMin x r join x l Tip = insertMax x l join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz) | delta*sizeL < sizeR = balanceL z (join x l lz) rz | delta*sizeR < sizeL = balanceR y ly (join x ry r) | otherwise = bin x l r -- insertMin and insertMax don't perform potentially expensive comparisons. insertMax,insertMin :: a -> Set a -> Set a insertMax x t = case t of Tip -> singleton x Bin _ y l r -> balanceR y l (insertMax x r) insertMin x t = case t of Tip -> singleton x Bin _ y l r -> balanceL y (insertMin x l) r {-------------------------------------------------------------------- [merge l r]: merges two trees. --------------------------------------------------------------------} merge :: Set a -> Set a -> Set a merge Tip r = r merge l Tip = l merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry) | delta*sizeL < sizeR = balanceL y (merge l ly) ry | delta*sizeR < sizeL = balanceR x lx (merge rx r) | otherwise = glue l r {-------------------------------------------------------------------- [glue l r]: glues two trees together. Assumes that [l] and [r] are already balanced with respect to each other. --------------------------------------------------------------------} glue :: Set a -> Set a -> Set a glue Tip r = r glue l Tip = l glue l r | size l > size r = let (m,l') = deleteFindMax l in balanceR m l' r | otherwise = let (m,r') = deleteFindMin r in balanceL m l r' -- | /O(log n)/. Delete and find the minimal element. -- -- > deleteFindMin set = (findMin set, deleteMin set) deleteFindMin :: Set a -> (a,Set a) deleteFindMin t = case t of Bin _ x Tip r -> (x,r) Bin _ x l r -> let (xm,l') = deleteFindMin l in (xm,balanceR x l' r) Tip -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip) -- | /O(log n)/. Delete and find the maximal element. -- -- > deleteFindMax set = (findMax set, deleteMax set) deleteFindMax :: Set a -> (a,Set a) deleteFindMax t = case t of Bin _ x l Tip -> (x,l) Bin _ x l r -> let (xm,r') = deleteFindMax r in (xm,balanceL x l r') Tip -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip) -- | /O(log n)/. Retrieves the minimal key of the set, and the set -- stripped of that element, or 'Nothing' if passed an empty set. minView :: Set a -> Maybe (a, Set a) minView Tip = Nothing minView x = Just (deleteFindMin x) -- | /O(log n)/. Retrieves the maximal key of the set, and the set -- stripped of that element, or 'Nothing' if passed an empty set. maxView :: Set a -> Maybe (a, Set a) maxView Tip = Nothing maxView x = Just (deleteFindMax x) {-------------------------------------------------------------------- [balance x l r] balances two trees with value x. The sizes of the trees should balance after decreasing the size of one of them. (a rotation). [delta] is the maximal relative difference between the sizes of two trees, it corresponds with the [w] in Adams' paper. [ratio] is the ratio between an outer and inner sibling of the heavier subtree in an unbalanced setting. It determines whether a double or single rotation should be performed to restore balance. It is correspondes with the inverse of $\alpha$ in Adam's article. Note that according to the Adam's paper: - [delta] should be larger than 4.646 with a [ratio] of 2. - [delta] should be larger than 3.745 with a [ratio] of 1.534. But the Adam's paper is errorneous: - it can be proved that for delta=2 and delta>=5 there does not exist any ratio that would work - delta=4.5 and ratio=2 does not work That leaves two reasonable variants, delta=3 and delta=4, both with ratio=2. - A lower [delta] leads to a more 'perfectly' balanced tree. - A higher [delta] performs less rebalancing. In the benchmarks, delta=3 is faster on insert operations, and delta=4 has slightly better deletes. As the insert speedup is larger, we currently use delta=3. --------------------------------------------------------------------} delta,ratio :: Int delta = 3 ratio = 2 -- The balance function is equivalent to the following: -- -- balance :: a -> Set a -> Set a -> Set a -- balance x l r -- | sizeL + sizeR <= 1 = Bin sizeX x l r -- | sizeR > delta*sizeL = rotateL x l r -- | sizeL > delta*sizeR = rotateR x l r -- | otherwise = Bin sizeX x l r -- where -- sizeL = size l -- sizeR = size r -- sizeX = sizeL + sizeR + 1 -- -- rotateL :: a -> Set a -> Set a -> Set a -- rotateL x l r@(Bin _ _ ly ry) | size ly < ratio*size ry = singleL x l r -- | otherwise = doubleL x l r -- rotateR :: a -> Set a -> Set a -> Set a -- rotateR x l@(Bin _ _ ly ry) r | size ry < ratio*size ly = singleR x l r -- | otherwise = doubleR x l r -- -- singleL, singleR :: a -> Set a -> Set a -> Set a -- singleL x1 t1 (Bin _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3 -- singleR x1 (Bin _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3) -- -- doubleL, doubleR :: a -> Set a -> Set a -> Set a -- doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4) -- doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4) -- -- It is only written in such a way that every node is pattern-matched only once. -- -- Only balanceL and balanceR are needed at the moment, so balance is not here anymore. -- In case it is needed, it can be found in Data.Map. -- Functions balanceL and balanceR are specialised versions of balance. -- balanceL only checks whether the left subtree is too big, -- balanceR only checks whether the right subtree is too big. -- balanceL is called when left subtree might have been inserted to or when -- right subtree might have been deleted from. balanceL :: a -> Set a -> Set a -> Set a balanceL x l r = case r of Tip -> case l of Tip -> Bin 1 x Tip Tip (Bin _ _ Tip Tip) -> Bin 2 x l Tip (Bin _ lx Tip (Bin _ lrx _ _)) -> Bin 3 lrx (Bin 1 lx Tip Tip) (Bin 1 x Tip Tip) (Bin _ lx ll@(Bin _ _ _ _) Tip) -> Bin 3 lx ll (Bin 1 x Tip Tip) (Bin ls lx ll@(Bin lls _ _ _) lr@(Bin lrs lrx lrl lrr)) | lrs < ratio*lls -> Bin (1+ls) lx ll (Bin (1+lrs) x lr Tip) | otherwise -> Bin (1+ls) lrx (Bin (1+lls+size lrl) lx ll lrl) (Bin (1+size lrr) x lrr Tip) (Bin rs _ _ _) -> case l of Tip -> Bin (1+rs) x Tip r (Bin ls lx ll lr) | ls > delta*rs -> case (ll, lr) of (Bin lls _ _ _, Bin lrs lrx lrl lrr) | lrs < ratio*lls -> Bin (1+ls+rs) lx ll (Bin (1+rs+lrs) x lr r) | otherwise -> Bin (1+ls+rs) lrx (Bin (1+lls+size lrl) lx ll lrl) (Bin (1+rs+size lrr) x lrr r) (_, _) -> error "Failure in Data.Map.balanceL" | otherwise -> Bin (1+ls+rs) x l r {-# NOINLINE balanceL #-} -- balanceR is called when right subtree might have been inserted to or when -- left subtree might have been deleted from. balanceR :: a -> Set a -> Set a -> Set a balanceR x l r = case l of Tip -> case r of Tip -> Bin 1 x Tip Tip (Bin _ _ Tip Tip) -> Bin 2 x Tip r (Bin _ rx Tip rr@(Bin _ _ _ _)) -> Bin 3 rx (Bin 1 x Tip Tip) rr (Bin _ rx (Bin _ rlx _ _) Tip) -> Bin 3 rlx (Bin 1 x Tip Tip) (Bin 1 rx Tip Tip) (Bin rs rx rl@(Bin rls rlx rll rlr) rr@(Bin rrs _ _ _)) | rls < ratio*rrs -> Bin (1+rs) rx (Bin (1+rls) x Tip rl) rr | otherwise -> Bin (1+rs) rlx (Bin (1+size rll) x Tip rll) (Bin (1+rrs+size rlr) rx rlr rr) (Bin ls _ _ _) -> case r of Tip -> Bin (1+ls) x l Tip (Bin rs rx rl rr) | rs > delta*ls -> case (rl, rr) of (Bin rls rlx rll rlr, Bin rrs _ _ _) | rls < ratio*rrs -> Bin (1+ls+rs) rx (Bin (1+ls+rls) x l rl) rr | otherwise -> Bin (1+ls+rs) rlx (Bin (1+ls+size rll) x l rll) (Bin (1+rrs+size rlr) rx rlr rr) (_, _) -> error "Failure in Data.Map.balanceR" | otherwise -> Bin (1+ls+rs) x l r {-# NOINLINE balanceR #-} {-------------------------------------------------------------------- The bin constructor maintains the size of the tree --------------------------------------------------------------------} bin :: a -> Set a -> Set a -> Set a bin x l r = Bin (size l + size r + 1) x l r {-# INLINE bin #-} {-------------------------------------------------------------------- Utilities --------------------------------------------------------------------} foldlStrict :: (a -> b -> a) -> a -> [b] -> a foldlStrict f = go where go z [] = z go z (x:xs) = let z' = f z x in z' `seq` go z' xs {-# INLINE foldlStrict #-} {-------------------------------------------------------------------- Debugging --------------------------------------------------------------------} -- | /O(n)/. Show the tree that implements the set. The tree is shown -- in a compressed, hanging format. showTree :: Show a => Set a -> String showTree s = showTreeWith True False s {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows the tree that implements the set. If @hang@ is @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If @wide@ is 'True', an extra wide version is shown. > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5] > 4 > +--2 > | +--1 > | +--3 > +--5 > > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5] > 4 > | > +--2 > | | > | +--1 > | | > | +--3 > | > +--5 > > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5] > +--5 > | > 4 > | > | +--3 > | | > +--2 > | > +--1 -} showTreeWith :: Show a => Bool -> Bool -> Set a -> String showTreeWith hang wide t | hang = (showsTreeHang wide [] t) "" | otherwise = (showsTree wide [] [] t) "" showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS showsTree wide lbars rbars t = case t of Tip -> showsBars lbars . showString "|\n" Bin _ x Tip Tip -> showsBars lbars . shows x . showString "\n" Bin _ x l r -> showsTree wide (withBar rbars) (withEmpty rbars) r . showWide wide rbars . showsBars lbars . shows x . showString "\n" . showWide wide lbars . showsTree wide (withEmpty lbars) (withBar lbars) l showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS showsTreeHang wide bars t = case t of Tip -> showsBars bars . showString "|\n" Bin _ x Tip Tip -> showsBars bars . shows x . showString "\n" Bin _ x l r -> showsBars bars . shows x . showString "\n" . showWide wide bars . showsTreeHang wide (withBar bars) l . showWide wide bars . showsTreeHang wide (withEmpty bars) r showWide :: Bool -> [String] -> String -> String showWide wide bars | wide = showString (concat (reverse bars)) . showString "|\n" | otherwise = id showsBars :: [String] -> ShowS showsBars bars = case bars of [] -> id _ -> showString (concat (reverse (tail bars))) . showString node node :: String node = "+--" withBar, withEmpty :: [String] -> [String] withBar bars = "| ":bars withEmpty bars = " ":bars {-------------------------------------------------------------------- Assertions --------------------------------------------------------------------} -- | /O(n)/. Test if the internal set structure is valid. valid :: Ord a => Set a -> Bool valid t = balanced t && ordered t && validsize t ordered :: Ord a => Set a -> Bool ordered t = bounded (const True) (const True) t where bounded lo hi t' = case t' of Tip -> True Bin _ x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r balanced :: Set a -> Bool balanced t = case t of Tip -> True Bin _ _ l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) && balanced l && balanced r validsize :: Set a -> Bool validsize t = (realsize t == Just (size t)) where realsize t' = case t' of Tip -> Just 0 Bin sz _ l r -> case (realsize l,realsize r) of (Just n,Just m) | n+m+1 == sz -> Just sz _ -> Nothing