{-# LANGUAGE CPP #-} {-# LANGUAGE MagicHash #-} -- License: The Glasgow Haskell Compiler License -- -- Copyright 2004, The University Court of the University of Glasgow. -- All rights reserved. -- -- Redistribution and use in source and binary forms, with or without -- modification, are permitted provided that the following conditions are met: -- -- - Redistributions of source code must retain the above copyright notice, -- this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright notice, -- this list of conditions and the following disclaimer in the documentation -- and/or other materials provided with the distribution. -- -- - Neither name of the University nor the names of its contributors may be -- used to endorse or promote products derived from this software without -- specific prior written permission. -- -- THIS SOFTWARE IS PROVIDED BY THE UNIVERSITY COURT OF THE UNIVERSITY OF -- GLASGOW AND THE CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, -- INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND -- FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE -- UNIVERSITY COURT OF THE UNIVERSITY OF GLASGOW OR THE CONTRIBUTORS BE LIABLE -- FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL -- DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR -- SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER -- CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT -- LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY -- OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH -- DAMAGE. ----------------------------------------------------------------------------- -- | -- Module : Data.IntSet -- Copyright : (c) Daan Leijen 2002 -- License : BSD-style -- Maintainer : libraries@haskell.org -- Stability : provisional -- Portability : portable -- -- An efficient implementation of integer sets. -- -- Since many function names (but not the type name) clash with -- "Prelude" names, this module is usually imported @qualified@, e.g. -- -- > import Data.IntSet (IntSet) -- > import qualified Data.IntSet as IntSet -- -- The implementation is based on /big-endian patricia trees/. This data -- structure performs especially well on binary operations like 'union' -- and 'intersection'. However, my benchmarks show that it is also -- (much) faster on insertions and deletions when compared to a generic -- size-balanced set implementation (see "Data.Set"). -- -- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\", -- Workshop on ML, September 1998, pages 77-86, -- -- -- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve -- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4), -- October 1968, pages 514-534. -- -- Many operations have a worst-case complexity of /O(min(n,W))/. -- This means that the operation can become linear in the number of -- elements with a maximum of /W/ -- the number of bits in an 'Int' -- (32 or 64). ----------------------------------------------------------------------------- module GraphHammer.IntSet ( -- * Set type -- Exported with details for some faster operations in IntMap nearby (sergueyz). IntSet(..) -- instance Eq,Show -- * Operators , (\\) -- * Query , null , size , member , notMember , isSubsetOf , isProperSubsetOf -- * Construction , empty , singleton , insert , delete -- * Combine , union, unions , difference , intersection -- * Filter , filter , partition , split , splitMember -- * Min\/Max , findMin , findMax , deleteMin , deleteMax , deleteFindMin , deleteFindMax , maxView , minView -- * Map , map -- * Fold , fold -- * Conversion -- ** List , elems , toList , fromList -- ** Ordered list , toAscList , fromAscList , fromDistinctAscList -- * Debugging , showTree , showTreeWith ) where import Prelude hiding (lookup,filter,foldr,foldl,null,map) import Data.Bits import qualified Data.List as List import Data.Monoid (Monoid(..)) import Data.Maybe (fromMaybe) import Data.Int import Data.Word import Text.Read infixl 9 \\{-This comment teaches CPP correct behaviour -} -- A "Nat" is a natural machine word (an unsigned Int) type Nat = Word32 natFromInt :: Int32 -> Nat natFromInt i = fromIntegral i intFromNat :: Nat -> Int32 intFromNat w = fromIntegral w shiftRL :: Nat -> Int -> Nat shiftRL x i = shiftR x i {-------------------------------------------------------------------- Operators --------------------------------------------------------------------} -- | /O(n+m)/. See 'difference'. (\\) :: IntSet -> IntSet -> IntSet m1 \\ m2 = difference m1 m2 {-------------------------------------------------------------------- Types --------------------------------------------------------------------} -- | A set of integers. data IntSet = Nil | Tip {-# UNPACK #-} !Key | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet -- Invariant: Nil is never found as a child of Bin. -- Invariant: The Mask is a power of 2. It is the largest bit position at which -- two elements of the set differ. -- Invariant: Prefix is the common high-order bits that all elements share to -- the left of the Mask bit. -- Invariant: In Bin prefix mask left right, left consists of the elements that -- don't have the mask bit set; right is all the elements that do. type Key = Int32 type Prefix = Int32 type Mask = Int32 instance Monoid IntSet where mempty = empty mappend = union mconcat = unions {-------------------------------------------------------------------- Query --------------------------------------------------------------------} -- | /O(1)/. Is the set empty? null :: IntSet -> Bool null Nil = True null _ = False -- | /O(n)/. Cardinality of the set. size :: IntSet -> Int size t = case t of Bin _ _ l r -> size l + size r Tip _ -> 1 Nil -> 0 -- | /O(min(n,W))/. Is the value a member of the set? member :: Key -> IntSet -> Bool member x t = case t of Bin p m l r | nomatch x p m -> False | zero x m -> member x l | otherwise -> member x r Tip y -> (x==y) Nil -> False -- | /O(min(n,W))/. Is the element not in the set? notMember :: Key -> IntSet -> Bool notMember k = not . member k -- 'lookup' is used by 'intersection' for left-biasing lookup :: Key -> IntSet -> Maybe Key lookup k t = let nk = natFromInt k in seq nk (lookupN nk t) lookupN :: Nat -> IntSet -> Maybe Key lookupN k t = case t of Bin _ m l r | zeroN k (natFromInt m) -> lookupN k l | otherwise -> lookupN k r Tip kx | (k == natFromInt kx) -> Just kx | otherwise -> Nothing Nil -> Nothing {-------------------------------------------------------------------- Construction --------------------------------------------------------------------} -- | /O(1)/. The empty set. empty :: IntSet empty = Nil -- | /O(1)/. A set of one element. singleton :: Key -> IntSet singleton x = Tip x {-------------------------------------------------------------------- Insert --------------------------------------------------------------------} -- | /O(min(n,W))/. Add a value to the set. When the value is already -- an element of the set, it is replaced by the new one, ie. 'insert' -- is left-biased. insert :: Key -> IntSet -> IntSet insert x t = case t of Bin p m l r | nomatch x p m -> join x (Tip x) p t | zero x m -> Bin p m (insert x l) r | otherwise -> Bin p m l (insert x r) Tip y | x==y -> Tip x | otherwise -> join x (Tip x) y t Nil -> Tip x -- right-biased insertion, used by 'union' insertR :: Key -> IntSet -> IntSet insertR x t = case t of Bin p m l r | nomatch x p m -> join x (Tip x) p t | zero x m -> Bin p m (insert x l) r | otherwise -> Bin p m l (insert x r) Tip y | x==y -> t | otherwise -> join x (Tip x) y t Nil -> Tip x -- | /O(min(n,W))/. Delete a value in the set. Returns the -- original set when the value was not present. delete :: Key -> IntSet -> IntSet delete x t = case t of Bin p m l r | nomatch x p m -> t | zero x m -> bin p m (delete x l) r | otherwise -> bin p m l (delete x r) Tip y | x==y -> Nil | otherwise -> t Nil -> Nil {-------------------------------------------------------------------- Union --------------------------------------------------------------------} -- | The union of a list of sets. unions :: [IntSet] -> IntSet unions xs = foldlStrict union empty xs -- | /O(n+m)/. The union of two sets. union :: IntSet -> IntSet -> IntSet union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2) | shorter m1 m2 = union1 | shorter m2 m1 = union2 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2) | otherwise = join p1 t1 p2 t2 where union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1 | otherwise = Bin p1 m1 l1 (union r1 t2) union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2 | otherwise = Bin p2 m2 l2 (union t1 r2) union (Tip x) t = insert x t union t (Tip x) = insertR x t -- right bias union Nil t = t union t Nil = t {-------------------------------------------------------------------- Difference --------------------------------------------------------------------} -- | /O(n+m)/. Difference between two sets. difference :: IntSet -> IntSet -> IntSet difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2) | shorter m1 m2 = difference1 | shorter m2 m1 = difference2 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2) | otherwise = t1 where difference1 | nomatch p2 p1 m1 = t1 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1 | otherwise = bin p1 m1 l1 (difference r1 t2) difference2 | nomatch p1 p2 m2 = t1 | zero p1 m2 = difference t1 l2 | otherwise = difference t1 r2 difference t1@(Tip x) t2 | member x t2 = Nil | otherwise = t1 difference Nil _ = Nil difference t (Tip x) = delete x t difference t Nil = t {-------------------------------------------------------------------- Intersection --------------------------------------------------------------------} -- | /O(n+m)/. The intersection of two sets. intersection :: IntSet -> IntSet -> IntSet intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2) | shorter m1 m2 = intersection1 | shorter m2 m1 = intersection2 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2) | otherwise = Nil where intersection1 | nomatch p2 p1 m1 = Nil | zero p2 m1 = intersection l1 t2 | otherwise = intersection r1 t2 intersection2 | nomatch p1 p2 m2 = Nil | zero p1 m2 = intersection t1 l2 | otherwise = intersection t1 r2 intersection t1@(Tip x) t2 | member x t2 = t1 | otherwise = Nil intersection t (Tip x) = case lookup x t of Just y -> Tip y Nothing -> Nil intersection Nil _ = Nil intersection _ Nil = Nil {-------------------------------------------------------------------- Subset --------------------------------------------------------------------} -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal). isProperSubsetOf :: IntSet -> IntSet -> Bool isProperSubsetOf t1 t2 = case subsetCmp t1 t2 of LT -> True _ -> False subsetCmp :: IntSet -> IntSet -> Ordering subsetCmp t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2) | shorter m1 m2 = GT | shorter m2 m1 = case subsetCmpLt of GT -> GT _ -> LT | p1 == p2 = subsetCmpEq | otherwise = GT -- disjoint where subsetCmpLt | nomatch p1 p2 m2 = GT | zero p1 m2 = subsetCmp t1 l2 | otherwise = subsetCmp t1 r2 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of (GT,_ ) -> GT (_ ,GT) -> GT (EQ,EQ) -> EQ _ -> LT subsetCmp (Bin _ _ _ _) _ = GT subsetCmp (Tip x) (Tip y) | x==y = EQ | otherwise = GT -- disjoint subsetCmp (Tip x) t | member x t = LT | otherwise = GT -- disjoint subsetCmp Nil Nil = EQ subsetCmp Nil _ = LT -- | /O(n+m)/. Is this a subset? -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@. isSubsetOf :: IntSet -> IntSet -> Bool isSubsetOf t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2) | shorter m1 m2 = False | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2 else isSubsetOf t1 r2) | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2 isSubsetOf (Bin _ _ _ _) _ = False isSubsetOf (Tip x) t = member x t isSubsetOf Nil _ = True {-------------------------------------------------------------------- Filter --------------------------------------------------------------------} -- | /O(n)/. Filter all elements that satisfy some predicate. filter :: (Key -> Bool) -> IntSet -> IntSet filter predicate t = case t of Bin p m l r -> bin p m (filter predicate l) (filter predicate r) Tip x | predicate x -> t | otherwise -> Nil Nil -> Nil -- | /O(n)/. partition the set according to some predicate. partition :: (Key -> Bool) -> IntSet -> (IntSet,IntSet) partition predicate t = case t of Bin p m l r -> let (l1,l2) = partition predicate l (r1,r2) = partition predicate r in (bin p m l1 r1, bin p m l2 r2) Tip x | predicate x -> (t,Nil) | otherwise -> (Nil,t) Nil -> (Nil,Nil) -- | /O(min(n,W))/. 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 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5]) split :: Key -> IntSet -> (IntSet,IntSet) split x t = case t of Bin _ m l r | m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt) else let (lt,gt) = split' x r in (lt, union gt l) -- handle negative numbers. | otherwise -> split' x t Tip y | x>y -> (t,Nil) | x (Nil,t) | otherwise -> (Nil,Nil) Nil -> (Nil, Nil) split' :: Key -> IntSet -> (IntSet,IntSet) split' x t = case t of Bin p m l r | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r) else let (lt,gt) = split' x r in (union l lt,gt) | otherwise -> if x < p then (Nil, t) else (t, Nil) Tip y | x>y -> (t,Nil) | x (Nil,t) | otherwise -> (Nil,Nil) Nil -> (Nil,Nil) -- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot -- element was found in the original set. splitMember :: Key -> IntSet -> (IntSet,Bool,IntSet) splitMember x t = case t of Bin _ m l r | m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt) else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l) -- handle negative numbers. | otherwise -> splitMember' x t Tip y | x>y -> (t,False,Nil) | x (Nil,False,t) | otherwise -> (Nil,True,Nil) Nil -> (Nil,False,Nil) splitMember' :: Key -> IntSet -> (IntSet,Bool,IntSet) splitMember' x t = case t of Bin p m l r | match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r) else let (lt,found,gt) = splitMember x r in (union l lt,found,gt) | otherwise -> if x < p then (Nil, False, t) else (t, False, Nil) Tip y | x>y -> (t,False,Nil) | x (Nil,False,t) | otherwise -> (Nil,True,Nil) Nil -> (Nil,False,Nil) {---------------------------------------------------------------------- Min/Max ----------------------------------------------------------------------} -- | /O(min(n,W))/. Retrieves the maximal key of the set, and the set -- stripped of that element, or 'Nothing' if passed an empty set. maxView :: IntSet -> Maybe (Key, IntSet) maxView t = case t of Bin p m l r | m < 0 -> let (result,t') = maxViewUnsigned l in Just (result, bin p m t' r) Bin p m l r -> let (result,t') = maxViewUnsigned r in Just (result, bin p m l t') Tip y -> Just (y,Nil) Nil -> Nothing maxViewUnsigned :: IntSet -> (Key, IntSet) maxViewUnsigned t = case t of Bin p m l r -> let (result,t') = maxViewUnsigned r in (result, bin p m l t') Tip y -> (y, Nil) Nil -> error "maxViewUnsigned Nil" -- | /O(min(n,W))/. Retrieves the minimal key of the set, and the set -- stripped of that element, or 'Nothing' if passed an empty set. minView :: IntSet -> Maybe (Key, IntSet) minView t = case t of Bin p m l r | m < 0 -> let (result,t') = minViewUnsigned r in Just (result, bin p m l t') Bin p m l r -> let (result,t') = minViewUnsigned l in Just (result, bin p m t' r) Tip y -> Just (y, Nil) Nil -> Nothing minViewUnsigned :: IntSet -> (Key, IntSet) minViewUnsigned t = case t of Bin p m l r -> let (result,t') = minViewUnsigned l in (result, bin p m t' r) Tip y -> (y, Nil) Nil -> error "minViewUnsigned Nil" -- | /O(min(n,W))/. Delete and find the minimal element. -- -- > deleteFindMin set = (findMin set, deleteMin set) deleteFindMin :: IntSet -> (Key, IntSet) deleteFindMin = fromMaybe (error "deleteFindMin: empty set has no minimal element") . minView -- | /O(min(n,W))/. Delete and find the maximal element. -- -- > deleteFindMax set = (findMax set, deleteMax set) deleteFindMax :: IntSet -> (Key, IntSet) deleteFindMax = fromMaybe (error "deleteFindMax: empty set has no maximal element") . maxView -- | /O(min(n,W))/. The minimal element of the set. findMin :: IntSet -> Key findMin Nil = error "findMin: empty set has no minimal element" findMin (Tip x) = x findMin (Bin _ m l r) | m < 0 = find r | otherwise = find l where find (Tip x) = x find (Bin _ _ l' _) = find l' find Nil = error "findMin Nil" -- | /O(min(n,W))/. The maximal element of a set. findMax :: IntSet -> Key findMax Nil = error "findMax: empty set has no maximal element" findMax (Tip x) = x findMax (Bin _ m l r) | m < 0 = find l | otherwise = find r where find (Tip x) = x find (Bin _ _ _ r') = find r' find Nil = error "findMax Nil" -- | /O(min(n,W))/. Delete the minimal element. deleteMin :: IntSet -> IntSet deleteMin = maybe (error "deleteMin: empty set has no minimal element") snd . minView -- | /O(min(n,W))/. Delete the maximal element. deleteMax :: IntSet -> IntSet deleteMax = maybe (error "deleteMax: empty set has no maximal element") snd . maxView {---------------------------------------------------------------------- Map ----------------------------------------------------------------------} -- | /O(n*min(n,W))/. -- @'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 :: (Key->Key) -> IntSet -> IntSet map f = fromList . List.map f . toList {-------------------------------------------------------------------- Fold --------------------------------------------------------------------} -- | /O(n)/. Fold over the elements of a set in an unspecified order. -- -- > sum set == fold (+) 0 set -- > elems set == fold (:) [] set fold :: (Key -> b -> b) -> b -> IntSet -> b fold f z t = case t of Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r -- put negative numbers before. Bin _ _ _ _ -> foldr f z t Tip x -> f x z Nil -> z foldr :: (Key -> b -> b) -> b -> IntSet -> b foldr f z t = case t of Bin _ _ l r -> foldr f (foldr f z r) l Tip x -> f x z Nil -> z {-------------------------------------------------------------------- List variations --------------------------------------------------------------------} -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList) elems :: IntSet -> [Key] elems s = toList s {-------------------------------------------------------------------- Lists --------------------------------------------------------------------} -- | /O(n)/. Convert the set to a list of elements. toList :: IntSet -> [Key] toList t = fold (:) [] t -- | /O(n)/. Convert the set to an ascending list of elements. toAscList :: IntSet -> [Key] toAscList t = toList t -- | /O(n*min(n,W))/. Create a set from a list of integers. fromList :: [Key] -> IntSet fromList xs = foldlStrict ins empty xs where ins t x = insert x t -- | /O(n)/. Build a set from an ascending list of elements. -- /The precondition (input list is ascending) is not checked./ fromAscList :: [Key] -> IntSet fromAscList [] = Nil fromAscList (x0 : xs0) = fromDistinctAscList (combineEq x0 xs0) where combineEq x' [] = [x'] combineEq x' (x:xs) | x==x' = combineEq x' xs | otherwise = x' : combineEq x xs -- | /O(n)/. Build a set from an ascending list of distinct elements. -- /The precondition (input list is strictly ascending) is not checked./ fromDistinctAscList :: [Key] -> IntSet fromDistinctAscList [] = Nil fromDistinctAscList (z0 : zs0) = work z0 zs0 Nada where work x [] stk = finish x (Tip x) stk work x (z:zs) stk = reduce z zs (branchMask z x) x (Tip x) stk reduce z zs _ px tx Nada = work z zs (Push px tx Nada) reduce z zs m px tx stk@(Push py ty stk') = let mxy = branchMask px py pxy = mask px mxy in if shorter m mxy then reduce z zs m pxy (Bin pxy mxy ty tx) stk' else work z zs (Push px tx stk) finish _ t Nada = t finish px tx (Push py ty stk) = finish p (join py ty px tx) stk where m = branchMask px py p = mask px m data Stack = Push {-# UNPACK #-} !Prefix !IntSet !Stack | Nada {-------------------------------------------------------------------- Eq --------------------------------------------------------------------} instance Eq IntSet where t1 == t2 = equal t1 t2 t1 /= t2 = nequal t1 t2 equal :: IntSet -> IntSet -> Bool equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2) = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2) equal (Tip x) (Tip y) = (x==y) equal Nil Nil = True equal _ _ = False nequal :: IntSet -> IntSet -> Bool nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2) = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2) nequal (Tip x) (Tip y) = (x/=y) nequal Nil Nil = False nequal _ _ = True {-------------------------------------------------------------------- Ord --------------------------------------------------------------------} instance Ord IntSet where compare s1 s2 = compare (toAscList s1) (toAscList s2) -- tentative implementation. See if more efficient exists. {-------------------------------------------------------------------- Show --------------------------------------------------------------------} instance Show IntSet where showsPrec p xs = showParen (p > 10) $ showString "fromList " . shows (toList xs) {- XXX unused code showSet :: [Int] -> 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 IntSet 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 --------------------------------------------------------------------} {-------------------------------------------------------------------- Debugging --------------------------------------------------------------------} -- | /O(n)/. Show the tree that implements the set. The tree is shown -- in a compressed, hanging format. showTree :: IntSet -> 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. -} showTreeWith :: Bool -> Bool -> IntSet -> String showTreeWith hang wide t | hang = (showsTreeHang wide [] t) "" | otherwise = (showsTree wide [] [] t) "" showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS showsTree wide lbars rbars t = case t of Bin p m l r -> showsTree wide (withBar rbars) (withEmpty rbars) r . showWide wide rbars . showsBars lbars . showString (showBin p m) . showString "\n" . showWide wide lbars . showsTree wide (withEmpty lbars) (withBar lbars) l Tip x -> showsBars lbars . showString " " . shows x . showString "\n" Nil -> showsBars lbars . showString "|\n" showsTreeHang :: Bool -> [String] -> IntSet -> ShowS showsTreeHang wide bars t = case t of Bin p m l r -> showsBars bars . showString (showBin p m) . showString "\n" . showWide wide bars . showsTreeHang wide (withBar bars) l . showWide wide bars . showsTreeHang wide (withEmpty bars) r Tip x -> showsBars bars . showString " " . shows x . showString "\n" Nil -> showsBars bars . showString "|\n" showBin :: Prefix -> Mask -> String showBin _ _ = "*" -- ++ show (p,m) 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 {-------------------------------------------------------------------- Helpers --------------------------------------------------------------------} {-------------------------------------------------------------------- Join --------------------------------------------------------------------} join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet join p1 t1 p2 t2 | zero p1 m = Bin p m t1 t2 | otherwise = Bin p m t2 t1 where m = branchMask p1 p2 p = mask p1 m {-------------------------------------------------------------------- @bin@ assures that we never have empty trees within a tree. --------------------------------------------------------------------} bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet bin _ _ l Nil = l bin _ _ Nil r = r bin p m l r = Bin p m l r {-------------------------------------------------------------------- Endian independent bit twiddling --------------------------------------------------------------------} zero :: Key -> Mask -> Bool zero i m = (natFromInt i) .&. (natFromInt m) == 0 nomatch,match :: Key -> Prefix -> Mask -> Bool nomatch i p m = (mask i m) /= p match i p m = (mask i m) == p -- Suppose a is largest such that 2^a divides 2*m. -- Then mask i m is i with the low a bits zeroed out. mask :: Key -> Mask -> Prefix mask i m = maskW (natFromInt i) (natFromInt m) zeroN :: Nat -> Nat -> Bool zeroN i m = (i .&. m) == 0 {-------------------------------------------------------------------- Big endian operations --------------------------------------------------------------------} maskW :: Nat -> Nat -> Prefix maskW i m = intFromNat (i .&. (complement (m-1) `xor` m)) shorter :: Mask -> Mask -> Bool shorter m1 m2 = (natFromInt m1) > (natFromInt m2) branchMask :: Prefix -> Prefix -> Mask branchMask p1 p2 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2)) {---------------------------------------------------------------------- Finding the highest bit (mask) in a word [x] can be done efficiently in three ways: * convert to a floating point value and the mantissa tells us the [log2(x)] that corresponds with the highest bit position. The mantissa is retrieved either via the standard C function [frexp] or by some bit twiddling on IEEE compatible numbers (float). Note that one needs to use at least [double] precision for an accurate mantissa of 32 bit numbers. * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit). * use processor specific assembler instruction (asm). The most portable way would be [bit], but is it efficient enough? I have measured the cycle counts of the different methods on an AMD Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction: highestBitMask: method cycles -------------- frexp 200 float 33 bit 11 asm 12 highestBit: method cycles -------------- frexp 195 float 33 bit 11 asm 11 Wow, the bit twiddling is on today's RISC like machines even faster than a single CISC instruction (BSR)! ----------------------------------------------------------------------} {---------------------------------------------------------------------- [highestBitMask] returns a word where only the highest bit is set. It is found by first setting all bits in lower positions than the highest bit and than taking an exclusive or with the original value. Allthough the function may look expensive, GHC compiles this into excellent C code that subsequently compiled into highly efficient machine code. The algorithm is derived from Jorg Arndt's FXT library. ----------------------------------------------------------------------} highestBitMask :: Nat -> Nat highestBitMask x0 = case (x0 .|. shiftRL x0 1) of x1 -> case (x1 .|. shiftRL x1 2) of x2 -> case (x2 .|. shiftRL x2 4) of x3 -> case (x3 .|. shiftRL x3 8) of x4 -> case (x4 .|. shiftRL x4 16) of x5 -> case (x5 .|. shiftRL x5 32) of -- for 64 bit platforms x6 -> (x6 `xor` (shiftRL x6 1)) {-------------------------------------------------------------------- Utilities --------------------------------------------------------------------} foldlStrict :: (a -> b -> a) -> a -> [b] -> a foldlStrict f z xs = case xs of [] -> z (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx) {- {-------------------------------------------------------------------- Testing --------------------------------------------------------------------} testTree :: [Int] -> IntSet 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 Arbitrary IntSet where arbitrary = do{ xs <- arbitrary ; return (fromList xs) } {-------------------------------------------------------------------- Single, Insert, Delete --------------------------------------------------------------------} prop_Single :: Int -> Bool prop_Single x = (insert x empty == singleton x) prop_InsertDelete :: Int -> IntSet -> Property prop_InsertDelete k t = not (member k t) ==> delete k (insert k t) == t {-------------------------------------------------------------------- Union --------------------------------------------------------------------} prop_UnionInsert :: Int -> IntSet -> Bool prop_UnionInsert x t = union t (singleton x) == insert x t prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool prop_UnionAssoc t1 t2 t3 = union t1 (union t2 t3) == union (union t1 t2) t3 prop_UnionComm :: IntSet -> IntSet -> Bool prop_UnionComm t1 t2 = (union t1 t2 == union t2 t1) prop_Diff :: [Int] -> [Int] -> Bool prop_Diff xs ys = toAscList (difference (fromList xs) (fromList ys)) == List.sort ((List.\\) (nub xs) (nub ys)) prop_Int :: [Int] -> [Int] -> Bool prop_Int xs ys = toAscList (intersection (fromList xs) (fromList ys)) == List.sort (nub ((List.intersect) (xs) (ys))) {-------------------------------------------------------------------- Lists --------------------------------------------------------------------} prop_Ordered = forAll (choose (5,100)) $ \n -> let xs = concat [[i-n,i-n]|i<-[0..2*n :: Int]] in fromAscList xs == fromList xs prop_List :: [Int] -> Bool prop_List xs = (sort (nub xs) == toAscList (fromList xs)) {-------------------------------------------------------------------- Bin invariants --------------------------------------------------------------------} powersOf2 :: IntSet powersOf2 = fromList [2^i | i <- [0..63]] -- Check the invariant that the mask is a power of 2. prop_MaskPow2 :: IntSet -> Bool prop_MaskPow2 (Bin _ msk left right) = member msk powersOf2 && prop_MaskPow2 left && prop_MaskPow2 right prop_MaskPow2 _ = True -- Check that the prefix satisfies its invariant. prop_Prefix :: IntSet -> Bool prop_Prefix s@(Bin prefix msk left right) = all (\elem -> match elem prefix msk) (toList s) && prop_Prefix left && prop_Prefix right prop_Prefix _ = True -- Check that the left elements don't have the mask bit set, and the right -- ones do. prop_LeftRight :: IntSet -> Bool prop_LeftRight (Bin _ msk left right) = and [x .&. msk == 0 | x <- toList left] && and [x .&. msk == msk | x <- toList right] prop_LeftRight _ = True {-------------------------------------------------------------------- IntSet operations are like Set operations --------------------------------------------------------------------} toSet :: IntSet -> Set.Set Int toSet = Set.fromList . toList -- Check that IntSet.isProperSubsetOf is the same as Set.isProperSubsetOf. prop_isProperSubsetOf :: IntSet -> IntSet -> Bool prop_isProperSubsetOf a b = isProperSubsetOf a b == Set.isProperSubsetOf (toSet a) (toSet b) -- In the above test, isProperSubsetOf almost always returns False (since a -- random set is almost never a subset of another random set). So this second -- test checks the True case. prop_isProperSubsetOf2 :: IntSet -> IntSet -> Bool prop_isProperSubsetOf2 a b = isProperSubsetOf a c == (a /= c) where c = union a b -}