1 {-# OPTIONS -cpp -fglasgow-exts #-}
2 --------------------------------------------------------------------------------
3 {-| Module : Data.IntSet
4 Copyright : (c) Daan Leijen 2002
6 Maintainer : libraries@haskell.org
7 Stability : provisional
10 An efficient implementation of integer sets.
12 This module is intended to be imported @qualified@, to avoid name
13 clashes with Prelude functions. eg.
15 > import Data.IntSet as Set
17 The implementation is based on /big-endian patricia trees/. This data structure
18 performs especially well on binary operations like 'union' and 'intersection'. However,
19 my benchmarks show that it is also (much) faster on insertions and deletions when
20 compared to a generic size-balanced set implementation (see "Set").
22 * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
23 Workshop on ML, September 1998, pages 77--86, <http://www.cse.ogi.edu/~andy/pub/finite.htm>
25 * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve Information
26 Coded In Alphanumeric/\", Journal of the ACM, 15(4), October 1968, pages 514--534.
28 Many operations have a worst-case complexity of /O(min(n,W))/. This means that the
29 operation can become linear in the number of elements
30 with a maximum of /W/ -- the number of bits in an 'Int' (32 or 64).
32 ---------------------------------------------------------------------------------}
35 IntSet -- instance Eq,Show
87 import Prelude hiding (lookup,filter,foldr,foldl,null,map)
91 import qualified Data.List as List
97 import List (nub,sort)
102 #ifdef __GLASGOW_HASKELL__
103 {--------------------------------------------------------------------
104 GHC: use unboxing to get @shiftRL@ inlined.
105 --------------------------------------------------------------------}
106 #if __GLASGOW_HASKELL__ >= 503
108 import GHC.Exts ( Word(..), Int(..), shiftRL# )
111 import GlaExts ( Word(..), Int(..), shiftRL# )
114 infixl 9 \\{-This comment teaches CPP correct behaviour -}
118 natFromInt :: Int -> Nat
119 natFromInt i = fromIntegral i
121 intFromNat :: Nat -> Int
122 intFromNat w = fromIntegral w
124 shiftRL :: Nat -> Int -> Nat
125 shiftRL (W# x) (I# i)
129 {--------------------------------------------------------------------
131 * raises errors on boundary values when using 'fromIntegral'
132 but not with the deprecated 'fromInt/toInt'.
133 * Older Hugs doesn't define 'Word'.
134 * Newer Hugs defines 'Word' in the Prelude but no operations.
135 --------------------------------------------------------------------}
139 type Nat = Word32 -- illegal on 64-bit platforms!
141 natFromInt :: Int -> Nat
142 natFromInt i = fromInt i
144 intFromNat :: Nat -> Int
145 intFromNat w = toInt w
147 shiftRL :: Nat -> Int -> Nat
148 shiftRL x i = shiftR x i
151 {--------------------------------------------------------------------
153 * A "Nat" is a natural machine word (an unsigned Int)
154 --------------------------------------------------------------------}
160 natFromInt :: Int -> Nat
161 natFromInt i = fromIntegral i
163 intFromNat :: Nat -> Int
164 intFromNat w = fromIntegral w
166 shiftRL :: Nat -> Int -> Nat
167 shiftRL w i = shiftR w i
171 {--------------------------------------------------------------------
173 --------------------------------------------------------------------}
174 -- | /O(n+m)/. See 'difference'.
175 (\\) :: IntSet -> IntSet -> IntSet
176 m1 \\ m2 = difference m1 m2
178 {--------------------------------------------------------------------
180 --------------------------------------------------------------------}
181 -- | A set of integers.
183 | Tip {-# UNPACK #-} !Int
184 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
189 {--------------------------------------------------------------------
191 --------------------------------------------------------------------}
192 -- | /O(1)/. Is the set empty?
193 null :: IntSet -> Bool
197 -- | /O(n)/. Cardinality of the set.
198 size :: IntSet -> Int
201 Bin p m l r -> size l + size r
205 -- | /O(min(n,W))/. Is the value a member of the set?
206 member :: Int -> IntSet -> Bool
210 | nomatch x p m -> False
211 | zero x m -> member x l
212 | otherwise -> member x r
216 -- 'lookup' is used by 'intersection' for left-biasing
217 lookup :: Int -> IntSet -> Maybe Int
219 = let nk = natFromInt k in seq nk (lookupN nk t)
221 lookupN :: Nat -> IntSet -> Maybe Int
225 | zeroN k (natFromInt m) -> lookupN k l
226 | otherwise -> lookupN k r
228 | (k == natFromInt kx) -> Just kx
229 | otherwise -> Nothing
232 {--------------------------------------------------------------------
234 --------------------------------------------------------------------}
235 -- | /O(1)/. The empty set.
240 -- | /O(1)/. A set of one element.
241 singleton :: Int -> IntSet
245 {--------------------------------------------------------------------
247 --------------------------------------------------------------------}
248 -- | /O(min(n,W))/. Add a value to the set. When the value is already
249 -- an element of the set, it is replaced by the new one, ie. 'insert'
251 insert :: Int -> IntSet -> IntSet
255 | nomatch x p m -> join x (Tip x) p t
256 | zero x m -> Bin p m (insert x l) r
257 | otherwise -> Bin p m l (insert x r)
260 | otherwise -> join x (Tip x) y t
263 -- right-biased insertion, used by 'union'
264 insertR :: Int -> IntSet -> IntSet
268 | nomatch x p m -> join x (Tip x) p t
269 | zero x m -> Bin p m (insert x l) r
270 | otherwise -> Bin p m l (insert x r)
273 | otherwise -> join x (Tip x) y t
276 -- | /O(min(n,W))/. Delete a value in the set. Returns the
277 -- original set when the value was not present.
278 delete :: Int -> IntSet -> IntSet
283 | zero x m -> bin p m (delete x l) r
284 | otherwise -> bin p m l (delete x r)
291 {--------------------------------------------------------------------
293 --------------------------------------------------------------------}
294 -- | The union of a list of sets.
295 unions :: [IntSet] -> IntSet
297 = foldlStrict union empty xs
300 -- | /O(n+m)/. The union of two sets.
301 union :: IntSet -> IntSet -> IntSet
302 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
303 | shorter m1 m2 = union1
304 | shorter m2 m1 = union2
305 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
306 | otherwise = join p1 t1 p2 t2
308 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
309 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
310 | otherwise = Bin p1 m1 l1 (union r1 t2)
312 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
313 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
314 | otherwise = Bin p2 m2 l2 (union t1 r2)
316 union (Tip x) t = insert x t
317 union t (Tip x) = insertR x t -- right bias
322 {--------------------------------------------------------------------
324 --------------------------------------------------------------------}
325 -- | /O(n+m)/. Difference between two sets.
326 difference :: IntSet -> IntSet -> IntSet
327 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
328 | shorter m1 m2 = difference1
329 | shorter m2 m1 = difference2
330 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
333 difference1 | nomatch p2 p1 m1 = t1
334 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
335 | otherwise = bin p1 m1 l1 (difference r1 t2)
337 difference2 | nomatch p1 p2 m2 = t1
338 | zero p1 m2 = difference t1 l2
339 | otherwise = difference t1 r2
341 difference t1@(Tip x) t2
345 difference Nil t = Nil
346 difference t (Tip x) = delete x t
351 {--------------------------------------------------------------------
353 --------------------------------------------------------------------}
354 -- | /O(n+m)/. The intersection of two sets.
355 intersection :: IntSet -> IntSet -> IntSet
356 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
357 | shorter m1 m2 = intersection1
358 | shorter m2 m1 = intersection2
359 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
362 intersection1 | nomatch p2 p1 m1 = Nil
363 | zero p2 m1 = intersection l1 t2
364 | otherwise = intersection r1 t2
366 intersection2 | nomatch p1 p2 m2 = Nil
367 | zero p1 m2 = intersection t1 l2
368 | otherwise = intersection t1 r2
370 intersection t1@(Tip x) t2
373 intersection t (Tip x)
377 intersection Nil t = Nil
378 intersection t Nil = Nil
382 {--------------------------------------------------------------------
384 --------------------------------------------------------------------}
385 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
386 isProperSubsetOf :: IntSet -> IntSet -> Bool
387 isProperSubsetOf t1 t2
388 = case subsetCmp t1 t2 of
392 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
394 | shorter m2 m1 = subsetCmpLt
395 | p1 == p2 = subsetCmpEq
396 | otherwise = GT -- disjoint
398 subsetCmpLt | nomatch p1 p2 m2 = GT
399 | zero p1 m2 = subsetCmp t1 l2
400 | otherwise = subsetCmp t1 r2
401 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
407 subsetCmp (Bin p m l r) t = GT
408 subsetCmp (Tip x) (Tip y)
410 | otherwise = GT -- disjoint
413 | otherwise = GT -- disjoint
414 subsetCmp Nil Nil = EQ
417 -- | /O(n+m)/. Is this a subset?
418 -- @(s1 `isSubsetOf` s2)@ tells whether s1 is a subset of s2.
420 isSubsetOf :: IntSet -> IntSet -> Bool
421 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
422 | shorter m1 m2 = False
423 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
424 else isSubsetOf t1 r2)
425 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
426 isSubsetOf (Bin p m l r) t = False
427 isSubsetOf (Tip x) t = member x t
428 isSubsetOf Nil t = True
431 {--------------------------------------------------------------------
433 --------------------------------------------------------------------}
434 -- | /O(n)/. Filter all elements that satisfy some predicate.
435 filter :: (Int -> Bool) -> IntSet -> IntSet
439 -> bin p m (filter pred l) (filter pred r)
445 -- | /O(n)/. partition the set according to some predicate.
446 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
450 -> let (l1,l2) = partition pred l
451 (r1,r2) = partition pred r
452 in (bin p m l1 r1, bin p m l2 r2)
455 | otherwise -> (Nil,t)
459 -- | /O(log n)/. The expression (@split x set@) is a pair @(set1,set2)@
460 -- where all elements in @set1@ are lower than @x@ and all elements in
461 -- @set2@ larger than @x@.
463 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
464 split :: Int -> IntSet -> (IntSet,IntSet)
468 | zero x m -> let (lt,gt) = split x l in (lt,union gt r)
469 | otherwise -> let (lt,gt) = split x r in (union l lt,gt)
473 | otherwise -> (Nil,Nil)
476 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
477 -- element was found in the original set.
478 splitMember :: Int -> IntSet -> (Bool,IntSet,IntSet)
482 | zero x m -> let (found,lt,gt) = splitMember x l in (found,lt,union gt r)
483 | otherwise -> let (found,lt,gt) = splitMember x r in (found,union l lt,gt)
485 | x>y -> (False,t,Nil)
486 | x<y -> (False,Nil,t)
487 | otherwise -> (True,Nil,Nil)
488 Nil -> (False,Nil,Nil)
490 {----------------------------------------------------------------------
492 ----------------------------------------------------------------------}
494 -- | /O(n*min(n,W))/.
495 -- @map f s@ is the set obtained by applying @f@ to each element of @s@.
497 -- It's worth noting that the size of the result may be smaller if,
498 -- for some @(x,y)@, @x \/= y && f x == f y@
500 map :: (Int->Int) -> IntSet -> IntSet
501 map f = fromList . List.map f . toList
503 {--------------------------------------------------------------------
505 --------------------------------------------------------------------}
506 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
508 -- > sum set == fold (+) 0 set
509 -- > elems set == fold (:) [] set
510 fold :: (Int -> b -> b) -> b -> IntSet -> b
514 foldr :: (Int -> b -> b) -> b -> IntSet -> b
517 Bin p m l r -> foldr f (foldr f z r) l
521 {--------------------------------------------------------------------
523 --------------------------------------------------------------------}
524 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
525 elems :: IntSet -> [Int]
529 {--------------------------------------------------------------------
531 --------------------------------------------------------------------}
532 -- | /O(n)/. Convert the set to a list of elements.
533 toList :: IntSet -> [Int]
537 -- | /O(n)/. Convert the set to an ascending list of elements.
538 toAscList :: IntSet -> [Int]
540 = -- NOTE: the following algorithm only works for big-endian trees
541 let (pos,neg) = span (>=0) (foldr (:) [] t) in neg ++ pos
543 -- | /O(n*min(n,W))/. Create a set from a list of integers.
544 fromList :: [Int] -> IntSet
546 = foldlStrict ins empty xs
550 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
551 fromAscList :: [Int] -> IntSet
555 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
556 fromDistinctAscList :: [Int] -> IntSet
557 fromDistinctAscList xs
561 {--------------------------------------------------------------------
563 --------------------------------------------------------------------}
564 instance Eq IntSet where
565 t1 == t2 = equal t1 t2
566 t1 /= t2 = nequal t1 t2
568 equal :: IntSet -> IntSet -> Bool
569 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
570 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
571 equal (Tip x) (Tip y)
576 nequal :: IntSet -> IntSet -> Bool
577 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
578 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
579 nequal (Tip x) (Tip y)
581 nequal Nil Nil = False
584 {--------------------------------------------------------------------
586 --------------------------------------------------------------------}
588 instance Ord IntSet where
589 compare s1 s2 = compare (toAscList s1) (toAscList s2)
590 -- tentative implementation. See if more efficient exists.
592 {--------------------------------------------------------------------
594 --------------------------------------------------------------------}
596 instance Monoid IntSet where
601 {--------------------------------------------------------------------
603 --------------------------------------------------------------------}
604 instance Show IntSet where
605 showsPrec d s = showSet (toList s)
607 showSet :: [Int] -> ShowS
611 = showChar '{' . shows x . showTail xs
613 showTail [] = showChar '}'
614 showTail (x:xs) = showChar ',' . shows x . showTail xs
616 {--------------------------------------------------------------------
618 --------------------------------------------------------------------}
619 -- | /O(n)/. Show the tree that implements the set. The tree is shown
620 -- in a compressed, hanging format.
621 showTree :: IntSet -> String
623 = showTreeWith True False s
626 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
627 the tree that implements the set. If @hang@ is
628 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
629 @wide@ is true, an extra wide version is shown.
631 showTreeWith :: Bool -> Bool -> IntSet -> String
632 showTreeWith hang wide t
633 | hang = (showsTreeHang wide [] t) ""
634 | otherwise = (showsTree wide [] [] t) ""
636 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
637 showsTree wide lbars rbars t
640 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
641 showWide wide rbars .
642 showsBars lbars . showString (showBin p m) . showString "\n" .
643 showWide wide lbars .
644 showsTree wide (withEmpty lbars) (withBar lbars) l
646 -> showsBars lbars . showString " " . shows x . showString "\n"
647 Nil -> showsBars lbars . showString "|\n"
649 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
650 showsTreeHang wide bars t
653 -> showsBars bars . showString (showBin p m) . showString "\n" .
655 showsTreeHang wide (withBar bars) l .
657 showsTreeHang wide (withEmpty bars) r
659 -> showsBars bars . showString " " . shows x . showString "\n"
660 Nil -> showsBars bars . showString "|\n"
663 = "*" -- ++ show (p,m)
666 | wide = showString (concat (reverse bars)) . showString "|\n"
669 showsBars :: [String] -> ShowS
673 _ -> showString (concat (reverse (tail bars))) . showString node
676 withBar bars = "| ":bars
677 withEmpty bars = " ":bars
680 {--------------------------------------------------------------------
682 --------------------------------------------------------------------}
683 {--------------------------------------------------------------------
685 --------------------------------------------------------------------}
686 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
688 | zero p1 m = Bin p m t1 t2
689 | otherwise = Bin p m t2 t1
694 {--------------------------------------------------------------------
695 @bin@ assures that we never have empty trees within a tree.
696 --------------------------------------------------------------------}
697 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
700 bin p m l r = Bin p m l r
703 {--------------------------------------------------------------------
704 Endian independent bit twiddling
705 --------------------------------------------------------------------}
706 zero :: Int -> Mask -> Bool
708 = (natFromInt i) .&. (natFromInt m) == 0
710 nomatch,match :: Int -> Prefix -> Mask -> Bool
717 mask :: Int -> Mask -> Prefix
719 = maskW (natFromInt i) (natFromInt m)
721 zeroN :: Nat -> Nat -> Bool
722 zeroN i m = (i .&. m) == 0
724 {--------------------------------------------------------------------
725 Big endian operations
726 --------------------------------------------------------------------}
727 maskW :: Nat -> Nat -> Prefix
729 = intFromNat (i .&. (complement (m-1) `xor` m))
731 shorter :: Mask -> Mask -> Bool
733 = (natFromInt m1) > (natFromInt m2)
735 branchMask :: Prefix -> Prefix -> Mask
737 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
739 {----------------------------------------------------------------------
740 Finding the highest bit (mask) in a word [x] can be done efficiently in
742 * convert to a floating point value and the mantissa tells us the
743 [log2(x)] that corresponds with the highest bit position. The mantissa
744 is retrieved either via the standard C function [frexp] or by some bit
745 twiddling on IEEE compatible numbers (float). Note that one needs to
746 use at least [double] precision for an accurate mantissa of 32 bit
748 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
749 * use processor specific assembler instruction (asm).
751 The most portable way would be [bit], but is it efficient enough?
752 I have measured the cycle counts of the different methods on an AMD
753 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
755 highestBitMask: method cycles
762 highestBit: method cycles
769 Wow, the bit twiddling is on today's RISC like machines even faster
770 than a single CISC instruction (BSR)!
771 ----------------------------------------------------------------------}
773 {----------------------------------------------------------------------
774 [highestBitMask] returns a word where only the highest bit is set.
775 It is found by first setting all bits in lower positions than the
776 highest bit and than taking an exclusive or with the original value.
777 Allthough the function may look expensive, GHC compiles this into
778 excellent C code that subsequently compiled into highly efficient
779 machine code. The algorithm is derived from Jorg Arndt's FXT library.
780 ----------------------------------------------------------------------}
781 highestBitMask :: Nat -> Nat
783 = case (x .|. shiftRL x 1) of
784 x -> case (x .|. shiftRL x 2) of
785 x -> case (x .|. shiftRL x 4) of
786 x -> case (x .|. shiftRL x 8) of
787 x -> case (x .|. shiftRL x 16) of
788 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
789 x -> (x `xor` (shiftRL x 1))
792 {--------------------------------------------------------------------
794 --------------------------------------------------------------------}
798 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
802 {--------------------------------------------------------------------
804 --------------------------------------------------------------------}
805 testTree :: [Int] -> IntSet
806 testTree xs = fromList xs
807 test1 = testTree [1..20]
808 test2 = testTree [30,29..10]
809 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
811 {--------------------------------------------------------------------
813 --------------------------------------------------------------------}
818 { configMaxTest = 500
819 , configMaxFail = 5000
820 , configSize = \n -> (div n 2 + 3)
821 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
825 {--------------------------------------------------------------------
826 Arbitrary, reasonably balanced trees
827 --------------------------------------------------------------------}
828 instance Arbitrary IntSet where
829 arbitrary = do{ xs <- arbitrary
830 ; return (fromList xs)
834 {--------------------------------------------------------------------
835 Single, Insert, Delete
836 --------------------------------------------------------------------}
837 prop_Single :: Int -> Bool
839 = (insert x empty == singleton x)
841 prop_InsertDelete :: Int -> IntSet -> Property
842 prop_InsertDelete k t
843 = not (member k t) ==> delete k (insert k t) == t
846 {--------------------------------------------------------------------
848 --------------------------------------------------------------------}
849 prop_UnionInsert :: Int -> IntSet -> Bool
851 = union t (singleton x) == insert x t
853 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
854 prop_UnionAssoc t1 t2 t3
855 = union t1 (union t2 t3) == union (union t1 t2) t3
857 prop_UnionComm :: IntSet -> IntSet -> Bool
859 = (union t1 t2 == union t2 t1)
861 prop_Diff :: [Int] -> [Int] -> Bool
863 = toAscList (difference (fromList xs) (fromList ys))
864 == List.sort ((List.\\) (nub xs) (nub ys))
866 prop_Int :: [Int] -> [Int] -> Bool
868 = toAscList (intersection (fromList xs) (fromList ys))
869 == List.sort (nub ((List.intersect) (xs) (ys)))
871 {--------------------------------------------------------------------
873 --------------------------------------------------------------------}
875 = forAll (choose (5,100)) $ \n ->
877 in fromAscList xs == fromList xs
879 prop_List :: [Int] -> Bool
881 = (sort (nub xs) == toAscList (fromList xs))