X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=Data%2FSet.hs;h=9ec9e4ce1a519148dedc89e1a14240a2f309e918;hb=74bc2d04fdbae494bcf4839c4ec5e6ec1d0bf600;hp=f4dad32a48e5d0b88253f917653bb9674ac1feb1;hpb=3742ec445c4a37d4897313b50ab95310ea55844e;p=haskell-directory.git diff --git a/Data/Set.hs b/Data/Set.hs index f4dad32..9ec9e4c 100644 --- a/Data/Set.hs +++ b/Data/Set.hs @@ -1,96 +1,1140 @@ ----------------------------------------------------------------------------- -- | -- Module : Data.Set --- Copyright : (c) The University of Glasgow 2001 --- License : BSD-style (see the file libraries/core/LICENSE) --- +-- Copyright : (c) Daan Leijen 2002 +-- License : BSD-style -- Maintainer : libraries@haskell.org -- Stability : provisional -- Portability : portable -- --- This implementation of sets sits squarely upon Data.FiniteMap. +-- An efficient implementation of sets. +-- +-- Since many function names (but not the type name) clash with +-- "Prelude" names, this module is usually imported @qualified@, 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, +-- . -- +-- * 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. ----------------------------------------------------------------------------- -module Data.Set ( - Set, -- abstract, instance of: Eq +module Data.Set ( + -- * Set type + Set -- instance Eq,Ord,Show,Read,Data,Typeable + + -- * Operators + , (\\) + + -- * Query + , null + , size + , member + , notMember + , isSubsetOf + , isProperSubsetOf + + -- * Construction + , empty + , singleton + , insert + , delete + + -- * Combine + , union, unions + , difference + , intersection + + -- * Filter + , filter + , partition + , split + , splitMember + + -- * Map + , map + , mapMonotonic + + -- * Fold + , fold + + -- * Min\/Max + , findMin + , findMax + , deleteMin + , deleteMax + , deleteFindMin + , deleteFindMax + , maxView + , minView + + -- * Conversion + + -- ** List + , elems + , toList + , fromList + + -- ** Ordered list + , toAscList + , fromAscList + , fromDistinctAscList + + -- * Debugging + , showTree + , showTreeWith + , valid + ) where + +import Prelude hiding (filter,foldr,null,map) +import qualified Data.List as List +import Data.Monoid (Monoid(..)) +import Data.Typeable +import Data.Foldable (Foldable(foldMap)) + +{- +-- just for testing +import QuickCheck +import List (nub,sort) +import qualified List +-} + +#if __GLASGOW_HASKELL__ +import Text.Read +import Data.Generics.Basics +import Data.Generics.Instances +#endif + +{-------------------------------------------------------------------- + Operators +--------------------------------------------------------------------} +infixl 9 \\ -- + +-- | /O(n+m)/. See 'difference'. +(\\) :: Ord a => Set a -> Set a -> Set a +m1 \\ m2 = difference m1 m2 + +{-------------------------------------------------------------------- + Sets are size balanced trees +--------------------------------------------------------------------} +-- | A set of values @a@. +data Set a = Tip + | Bin {-# UNPACK #-} !Size a !(Set a) !(Set a) + +type Size = Int + +instance Ord a => Monoid (Set a) where + mempty = empty + mappend = union + mconcat = unions + +instance Foldable Set where + foldMap f Tip = mempty + foldMap f (Bin _s k l r) = foldMap f l `mappend` f k `mappend` 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 t + = case t of + Tip -> True + Bin sz x l r -> False + +-- | /O(1)/. The number of elements in the set. +size :: Set a -> Int +size t + = case t of + Tip -> 0 + Bin sz x l r -> sz + +-- | /O(log n)/. Is the element in the set? +member :: Ord a => a -> Set a -> Bool +member x t + = case t of + Tip -> False + Bin sz y l r + -> case compare x y of + LT -> member x l + GT -> member x r + EQ -> True + +-- | /O(log n)/. Is the element not in the set? +notMember :: Ord a => a -> Set a -> Bool +notMember x t = not $ member x t + +{-------------------------------------------------------------------- + Construction +--------------------------------------------------------------------} +-- | /O(1)/. The empty set. +empty :: Set a +empty + = Tip + +-- | /O(1)/. Create a singleton set. +singleton :: a -> Set a +singleton x + = Bin 1 x Tip Tip + +{-------------------------------------------------------------------- + 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. +insert :: Ord a => a -> Set a -> Set a +insert x t + = case t of + Tip -> singleton x + Bin sz y l r + -> case compare x y of + LT -> balance y (insert x l) r + GT -> balance y l (insert x r) + EQ -> Bin sz x l r + + +-- | /O(log n)/. Delete an element from a set. +delete :: Ord a => a -> Set a -> Set a +delete x t + = case t of + Tip -> Tip + Bin sz y l r + -> case compare x y of + LT -> balance y (delete x l) r + GT -> balance y l (delete x r) + EQ -> glue l r + +{-------------------------------------------------------------------- + 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) - emptySet, -- :: Set a - mkSet, -- :: Ord a => [a] -> Set a - setToList, -- :: Set a -> [a] - unitSet, -- :: a -> Set a - union, -- :: Ord a => Set a -> Set a -> Set a - unionManySets, -- :: Ord a => [Set a] -> Set a - minusSet, -- :: Ord a => Set a -> Set a -> Set a - mapSet, -- :: Ord a => (b -> a) -> Set b -> Set a - intersect, -- :: Ord a => Set a -> Set a -> Set a - addToSet, -- :: Ord a => Set a -> a -> Set a - delFromSet, -- :: Ord a => Set a -> a -> Set a +-- | /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) - elementOf, -- :: Ord a => a -> Set a -> Bool - isEmptySet, -- :: Set a -> Bool - - cardinality -- :: Set a -> Int - ) where +isSubsetOfX Tip t = True +isSubsetOfX t Tip = False +isSubsetOfX (Bin _ x l r) t + = found && isSubsetOfX l lt && isSubsetOfX r gt + where + (lt,found,gt) = splitMember x t -import Prelude -import Data.FiniteMap -import Data.Maybe +{-------------------------------------------------------------------- + Minimal, Maximal +--------------------------------------------------------------------} +-- | /O(log n)/. The minimal element of a set. +findMin :: Set a -> a +findMin (Bin _ x Tip r) = x +findMin (Bin _ x l r) = findMin l +findMin Tip = error "Set.findMin: empty set has no minimal element" --- This can't be a type synonym if you want to use constructor classes. -newtype Set a = MkSet (FiniteMap a ()) +-- | /O(log n)/. The maximal element of a set. +findMax :: Set a -> a +findMax (Bin _ x l Tip) = x +findMax (Bin _ x l r) = findMax r +findMax Tip = error "Set.findMax: empty set has no maximal element" -emptySet :: Set a -emptySet = MkSet emptyFM +-- | /O(log n)/. Delete the minimal element. +deleteMin :: Set a -> Set a +deleteMin (Bin _ x Tip r) = r +deleteMin (Bin _ x l r) = balance x (deleteMin l) r +deleteMin Tip = Tip -unitSet :: a -> Set a -unitSet x = MkSet (unitFM x ()) +-- | /O(log n)/. Delete the maximal element. +deleteMax :: Set a -> Set a +deleteMax (Bin _ x l Tip) = l +deleteMax (Bin _ x l r) = balance x l (deleteMax r) +deleteMax Tip = Tip -setToList :: Set a -> [a] -setToList (MkSet set) = keysFM set -mkSet :: Ord a => [a] -> Set a -mkSet xs = MkSet (listToFM [ (x, ()) | x <- xs]) +{-------------------------------------------------------------------- + Union. +--------------------------------------------------------------------} +-- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@). +unions :: Ord a => [Set a] -> Set a +unions ts + = foldlStrict union empty ts + +-- | /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 (MkSet set1) (MkSet set2) = MkSet (plusFM set1 set2) +union Tip t2 = t2 +union t1 Tip = t1 +union t1 t2 = hedgeUnion (const LT) (const GT) t1 t2 + +hedgeUnion cmplo cmphi t1 Tip + = t1 +hedgeUnion cmplo cmphi Tip (Bin _ x l r) + = join x (filterGt cmplo l) (filterLt cmphi r) +hedgeUnion cmplo cmphi (Bin _ x l r) t2 + = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2)) + (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2)) + where + cmpx y = compare x y + +{-------------------------------------------------------------------- + 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 t2 = Tip +difference t1 Tip = t1 +difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2 + +hedgeDiff cmplo cmphi Tip t + = Tip +hedgeDiff cmplo cmphi (Bin _ x l r) Tip + = join x (filterGt cmplo l) (filterLt cmphi r) +hedgeDiff cmplo cmphi t (Bin _ x l r) + = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l) + (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r) + where + cmpx y = compare x y + +{-------------------------------------------------------------------- + Intersection +--------------------------------------------------------------------} +-- | /O(n+m)/. The intersection of two sets. +-- Elements of the result come from the first set. +intersection :: Ord a => Set a -> Set a -> Set a +intersection Tip t = Tip +intersection t Tip = Tip +intersection t1@(Bin s1 x1 l1 r1) t2@(Bin s2 x2 l2 r2) = + if s1 >= s2 then + let (lt,found,gt) = splitLookup x2 t1 + tl = intersection lt l2 + tr = intersection gt r2 + in case found of + Just x -> join x tl tr + Nothing -> merge tl tr + else let (lt,found,gt) = splitMember x1 t2 + tl = intersection l1 lt + tr = intersection r1 gt + in if found then join x1 tl tr + else merge tl tr + +{-------------------------------------------------------------------- + Filter and partition +--------------------------------------------------------------------} +-- | /O(n)/. Filter all elements that satisfy the predicate. +filter :: Ord a => (a -> Bool) -> Set a -> Set a +filter p 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 :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a) +partition p Tip = (Tip,Tip) +partition p (Bin _ x l r) + | p x = (join x l1 r1,merge l2 r2) + | otherwise = (merge l1 r1,join x l2 r2) + where + (l1,l2) = partition p l + (r1,r2) = partition p r + +{---------------------------------------------------------------------- + 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 + +-- | /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 f Tip = Tip +mapMonotonic f (Bin sz x l r) = + Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r) + + +{-------------------------------------------------------------------- + Fold +--------------------------------------------------------------------} +-- | /O(n)/. Fold over the elements of a set in an unspecified order. +fold :: (a -> b -> b) -> b -> Set a -> b +fold f z s + = foldr f z s + +-- | /O(n)/. Post-order fold. +foldr :: (a -> b -> b) -> b -> Set a -> b +foldr f z Tip = z +foldr f z (Bin _ x l r) = foldr f (f x (foldr f z r)) l + +{-------------------------------------------------------------------- + List variations +--------------------------------------------------------------------} +-- | /O(n)/. The elements of a set. +elems :: Set a -> [a] +elems s + = toList s + +{-------------------------------------------------------------------- + Lists +--------------------------------------------------------------------} +-- | /O(n)/. Convert the set to a list of elements. +toList :: Set a -> [a] +toList s + = toAscList s + +-- | /O(n)/. Convert the set to an ascending list of elements. +toAscList :: Set a -> [a] +toAscList t + = foldr (:) [] t + + +-- | /O(n*log n)/. Create a set from a list of elements. +fromList :: Ord a => [a] -> Set a +fromList xs + = foldlStrict ins empty xs + where + ins t x = insert x t + +{-------------------------------------------------------------------- + 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 + + +-- | /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 + = build 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 build bushier trees. + build c 0 xs = c Tip xs + build c 5 xs = case xs of + (x1:x2:x3:x4:x5:xx) + -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx + build c n xs = seq nr $ build (buildR nr c) nl xs + where + nl = n `div` 2 + nr = n - nl - 1 + + buildR n c l (x:ys) = build (buildB l x c) n ys + buildB 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) + +showSet :: (Show a) => [a] -> ShowS +showSet [] + = showString "{}" +showSet (x:xs) + = showChar '{' . shows x . showTail xs + where + showTail [] = showChar '}' + showTail (x:xs) = showChar ',' . shows x . showTail 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") + +{-------------------------------------------------------------------- + Utility functions that return sub-ranges of the original + tree. Some functions take a comparison function as argument to + allow comparisons against infinite values. A function [cmplo x] + should be read as [compare lo x]. + + [trim cmplo cmphi t] A tree that is either empty or where [cmplo x == LT] + and [cmphi x == GT] for the value [x] of the root. + [filterGt cmp t] A tree where for all values [k]. [cmp k == LT] + [filterLt cmp t] A tree where for all values [k]. [cmp k == GT] + + [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. +--------------------------------------------------------------------} + +{-------------------------------------------------------------------- + [trim lo hi t] trims away all subtrees that surely contain no + values between the range [lo] to [hi]. The returned tree is either + empty or the key of the root is between @lo@ and @hi@. +--------------------------------------------------------------------} +trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a +trim cmplo cmphi Tip = Tip +trim cmplo cmphi t@(Bin sx x l r) + = case cmplo x of + LT -> case cmphi x of + GT -> t + le -> trim cmplo cmphi l + ge -> trim cmplo cmphi r + +trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a) +trimMemberLo lo cmphi Tip = (False,Tip) +trimMemberLo lo cmphi t@(Bin sx x l r) + = case compare lo x of + LT -> case cmphi x of + GT -> (member lo t, t) + le -> trimMemberLo lo cmphi l + GT -> trimMemberLo lo cmphi r + EQ -> (True,trim (compare lo) cmphi r) + + +{-------------------------------------------------------------------- + [filterGt x t] filter all values >[x] from tree [t] + [filterLt x t] filter all values <[x] from tree [t] +--------------------------------------------------------------------} +filterGt :: (a -> Ordering) -> Set a -> Set a +filterGt cmp Tip = Tip +filterGt cmp (Bin sx x l r) + = case cmp x of + LT -> join x (filterGt cmp l) r + GT -> filterGt cmp r + EQ -> r + +filterLt :: (a -> Ordering) -> Set a -> Set a +filterLt cmp Tip = Tip +filterLt cmp (Bin sx x l r) + = case cmp x of + LT -> filterLt cmp l + GT -> join x l (filterLt cmp r) + EQ -> l + + +{-------------------------------------------------------------------- + Split +--------------------------------------------------------------------} +-- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@ +-- where all elements in @set1@ are lower than @x@ and all elements in +-- @set2@ larger than @x@. @x@ is not found in neither @set1@ nor @set2@. +split :: Ord a => a -> Set a -> (Set a,Set a) +split x Tip = (Tip,Tip) +split x (Bin sy 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) + +-- | /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 x t = let (l,m,r) = splitLookup x t in + (l,maybe False (const True) m,r) + +-- | /O(log n)/. Performs a 'split' but also returns the pivot +-- element that was found in the original set. +splitLookup :: Ord a => a -> Set a -> (Set a,Maybe a,Set a) +splitLookup x Tip = (Tip,Nothing,Tip) +splitLookup x (Bin sy y l r) + = case compare x y of + LT -> let (lt,found,gt) = splitLookup x l in (lt,found,join y gt r) + GT -> let (lt,found,gt) = splitLookup x r in (join y l lt,found,gt) + EQ -> (l,Just y,r) + +{-------------------------------------------------------------------- + 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 = balance z (join x l lz) rz + | delta*sizeR <= sizeL = balance 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 sz y l r + -> balance y l (insertMax x r) + +insertMin x t + = case t of + Tip -> singleton x + Bin sz y l r + -> balance 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 = balance y (merge l ly) ry + | delta*sizeR <= sizeL = balance 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 balance m l' r + | otherwise = let (m,r') = deleteFindMin r in balance 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,balance 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,balance 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 from that element +-- @fail@s (in the monad) when passed an empty set. +minView :: Monad m => Set a -> m (a, Set a) +minView Tip = fail "Set.minView: empty set" +minView x = return (deleteFindMin x) + +-- | /O(log n)/. Retrieves the maximal key of the set, and the set stripped from that element +-- @fail@s (in the monad) when passed an empty set. +maxView :: Monad m => Set a -> m (a, Set a) +maxView Tip = fail "Set.maxView: empty set" +maxView x = return (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, + or equivalently, [1/delta] corresponds with the $\alpha$ + in Nievergelt's paper. Adams shows that [delta] should + be larger than 3.745 in order to garantee that the + rotations can always restore balance. + + [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: + - [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. + + - A lower [delta] leads to a more 'perfectly' balanced tree. + - A higher [delta] performs less rebalancing. + + - Balancing is automatic for random data and a balancing + scheme is only necessary to avoid pathological worst cases. + Almost any choice will do in practice + + - Allthough it seems that a rather large [delta] may perform better + than smaller one, measurements have shown that the smallest [delta] + of 4 is actually the fastest on a wide range of operations. It + especially improves performance on worst-case scenarios like + a sequence of ordered insertions. + + Note: in contrast to Adams' paper, we use a ratio of (at least) 2 + to decide whether a single or double rotation is needed. Allthough + he actually proves that this ratio is needed to maintain the + invariants, his implementation uses a (invalid) ratio of 1. + He is aware of the problem though since he has put a comment in his + original source code that he doesn't care about generating a + slightly inbalanced tree since it doesn't seem to matter in practice. + However (since we use quickcheck :-) we will stick to strictly balanced + trees. +--------------------------------------------------------------------} +delta,ratio :: Int +delta = 4 +ratio = 2 + +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 + +-- rotate +rotateL x l r@(Bin _ _ ly ry) + | size ly < ratio*size ry = singleL x l r + | otherwise = doubleL x l r + +rotateR x l@(Bin _ _ ly ry) r + | size ry < ratio*size ly = singleR x l r + | otherwise = doubleR x l r + +-- basic rotations +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 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) + + +{-------------------------------------------------------------------- + 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 + + +{-------------------------------------------------------------------- + Utilities +--------------------------------------------------------------------} +foldlStrict f z xs + = case xs of + [] -> z + (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx) + + +{-------------------------------------------------------------------- + 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 sz x Tip Tip + -> showsBars lbars . shows x . showString "\n" + Bin sz 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 sz x Tip Tip + -> showsBars bars . shows x . showString "\n" + Bin sz 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 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 = "+--" +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 t + = bounded (const True) (const True) t + where + bounded lo hi t + = case t of + Tip -> True + Bin sz x l r -> (lo x) && (hi x) && bounded lo (x) hi r + +balanced :: Set a -> Bool +balanced t + = case t of + Tip -> True + Bin sz x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) && + balanced l && balanced r + + +validsize t + = (realsize t == Just (size t)) + where + realsize t + = case t of + Tip -> Just 0 + Bin sz x l r -> case (realsize l,realsize r) of + (Just n,Just m) | n+m+1 == sz -> Just sz + other -> Nothing + +{- +{-------------------------------------------------------------------- + Testing +--------------------------------------------------------------------} +testTree :: [Int] -> Set Int +testTree xs = fromList xs +test1 = testTree [1..20] +test2 = testTree [30,29..10] +test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3] + +{-------------------------------------------------------------------- + QuickCheck +--------------------------------------------------------------------} +qcheck prop + = check config prop + where + config = Config + { configMaxTest = 500 + , configMaxFail = 5000 + , configSize = \n -> (div n 2 + 3) + , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ] + } + + +{-------------------------------------------------------------------- + Arbitrary, reasonably balanced trees +--------------------------------------------------------------------} +instance (Enum a) => Arbitrary (Set a) where + arbitrary = sized (arbtree 0 maxkey) + where maxkey = 10000 + +arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a) +arbtree lo hi n + | n <= 0 = return Tip + | lo >= hi = return Tip + | otherwise = do{ i <- choose (lo,hi) + ; m <- choose (1,30) + ; let (ml,mr) | m==(1::Int)= (1,2) + | m==2 = (2,1) + | m==3 = (1,1) + | otherwise = (2,2) + ; l <- arbtree lo (i-1) (n `div` ml) + ; r <- arbtree (i+1) hi (n `div` mr) + ; return (bin (toEnum i) l r) + } + + +{-------------------------------------------------------------------- + Valid tree's +--------------------------------------------------------------------} +forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property +forValid f + = forAll arbitrary $ \t -> +-- classify (balanced t) "balanced" $ + classify (size t == 0) "empty" $ + classify (size t > 0 && size t <= 10) "small" $ + classify (size t > 10 && size t <= 64) "medium" $ + classify (size t > 64) "large" $ + balanced t ==> f t + +forValidIntTree :: Testable a => (Set Int -> a) -> Property +forValidIntTree f + = forValid f + +forValidUnitTree :: Testable a => (Set Int -> a) -> Property +forValidUnitTree f + = forValid f + + +prop_Valid + = forValidUnitTree $ \t -> valid t + +{-------------------------------------------------------------------- + Single, Insert, Delete +--------------------------------------------------------------------} +prop_Single :: Int -> Bool +prop_Single x + = (insert x empty == singleton x) + +prop_InsertValid :: Int -> Property +prop_InsertValid k + = forValidUnitTree $ \t -> valid (insert k t) + +prop_InsertDelete :: Int -> Set Int -> Property +prop_InsertDelete k t + = not (member k t) ==> delete k (insert k t) == t + +prop_DeleteValid :: Int -> Property +prop_DeleteValid k + = forValidUnitTree $ \t -> + valid (delete k (insert k t)) + +{-------------------------------------------------------------------- + Balance +--------------------------------------------------------------------} +prop_Join :: Int -> Property +prop_Join x + = forValidUnitTree $ \t -> + let (l,r) = split x t + in valid (join x l r) + +prop_Merge :: Int -> Property +prop_Merge x + = forValidUnitTree $ \t -> + let (l,r) = split x t + in valid (merge l r) + -unionManySets :: Ord a => [Set a] -> Set a -unionManySets ss = foldr union emptySet ss +{-------------------------------------------------------------------- + Union +--------------------------------------------------------------------} +prop_UnionValid :: Property +prop_UnionValid + = forValidUnitTree $ \t1 -> + forValidUnitTree $ \t2 -> + valid (union t1 t2) -minusSet :: Ord a => Set a -> Set a -> Set a -minusSet (MkSet set1) (MkSet set2) = MkSet (minusFM set1 set2) +prop_UnionInsert :: Int -> Set Int -> Bool +prop_UnionInsert x t + = union t (singleton x) == insert x t -intersect :: Ord a => Set a -> Set a -> Set a -intersect (MkSet set1) (MkSet set2) = MkSet (intersectFM set1 set2) +prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool +prop_UnionAssoc t1 t2 t3 + = union t1 (union t2 t3) == union (union t1 t2) t3 -addToSet :: Ord a => Set a -> a -> Set a -addToSet (MkSet set) a = MkSet (addToFM set a ()) +prop_UnionComm :: Set Int -> Set Int -> Bool +prop_UnionComm t1 t2 + = (union t1 t2 == union t2 t1) -delFromSet :: Ord a => Set a -> a -> Set a -delFromSet (MkSet set) a = MkSet (delFromFM set a) -elementOf :: Ord a => a -> Set a -> Bool -elementOf x (MkSet set) = isJust (lookupFM set x) +prop_DiffValid + = forValidUnitTree $ \t1 -> + forValidUnitTree $ \t2 -> + valid (difference t1 t2) -isEmptySet :: Set a -> Bool -isEmptySet (MkSet set) = sizeFM set == 0 +prop_Diff :: [Int] -> [Int] -> Bool +prop_Diff xs ys + = toAscList (difference (fromList xs) (fromList ys)) + == List.sort ((List.\\) (nub xs) (nub ys)) -mapSet :: Ord a => (b -> a) -> Set b -> Set a -mapSet f (MkSet set) = MkSet (listToFM [ (f key, ()) | key <- keysFM set ]) +prop_IntValid + = forValidUnitTree $ \t1 -> + forValidUnitTree $ \t2 -> + valid (intersection t1 t2) -cardinality :: Set a -> Int -cardinality (MkSet set) = sizeFM set +prop_Int :: [Int] -> [Int] -> Bool +prop_Int xs ys + = toAscList (intersection (fromList xs) (fromList ys)) + == List.sort (nub ((List.intersect) (xs) (ys))) --- fair enough... -instance (Eq a) => Eq (Set a) where - (MkSet set_1) == (MkSet set_2) = set_1 == set_2 - (MkSet set_1) /= (MkSet set_2) = set_1 /= set_2 +{-------------------------------------------------------------------- + Lists +--------------------------------------------------------------------} +prop_Ordered + = forAll (choose (5,100)) $ \n -> + let xs = [0..n::Int] + in fromAscList xs == fromList xs --- but not so clear what the right thing to do is: -{- NO: -instance (Ord a) => Ord (Set a) where - (MkSet set_1) <= (MkSet set_2) = set_1 <= set_2 +prop_List :: [Int] -> Bool +prop_List xs + = (sort (nub xs) == toList (fromList xs)) -}