1 {-# OPTIONS -cpp -fglasgow-exts #-}
2 -----------------------------------------------------------------------------
4 -- Module : Data.IntSet
5 -- Copyright : (c) Daan Leijen 2002
7 -- Maintainer : libraries@haskell.org
8 -- Stability : provisional
9 -- Portability : portable
11 -- An efficient implementation of integer sets.
13 -- This module is intended to be imported @qualified@, to avoid name
14 -- clashes with "Prelude" functions. eg.
16 -- > import Data.IntSet as Set
18 -- The implementation is based on /big-endian patricia trees/. This data
19 -- structure performs especially well on binary operations like 'union'
20 -- and 'intersection'. However, my benchmarks show that it is also
21 -- (much) faster on insertions and deletions when compared to a generic
22 -- size-balanced set implementation (see "Data.Set").
24 -- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
25 -- Workshop on ML, September 1998, pages 77-86,
26 -- <http://www.cse.ogi.edu/~andy/pub/finite.htm>
28 -- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
29 -- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
30 -- October 1968, pages 514-534.
32 -- Many operations have a worst-case complexity of /O(min(n,W))/.
33 -- This means that the operation can become linear in the number of
34 -- elements with a maximum of /W/ -- the number of bits in an 'Int'
36 -----------------------------------------------------------------------------
40 IntSet -- instance Eq,Show
92 import Prelude hiding (lookup,filter,foldr,foldl,null,map)
96 import qualified Data.List as List
102 import List (nub,sort)
103 import qualified List
106 #if __GLASGOW_HASKELL__ >= 503
108 import GHC.Exts ( Word(..), Int(..), shiftRL# )
109 #elif __GLASGOW_HASKELL__
111 import GlaExts ( Word(..), Int(..), shiftRL# )
116 infixl 9 \\{-This comment teaches CPP correct behaviour -}
118 -- A "Nat" is a natural machine word (an unsigned Int)
121 natFromInt :: Int -> Nat
122 natFromInt i = fromIntegral i
124 intFromNat :: Nat -> Int
125 intFromNat w = fromIntegral w
127 shiftRL :: Nat -> Int -> Nat
128 #if __GLASGOW_HASKELL__
129 {--------------------------------------------------------------------
130 GHC: use unboxing to get @shiftRL@ inlined.
131 --------------------------------------------------------------------}
132 shiftRL (W# x) (I# i)
135 shiftRL x i = shiftR x i
138 {--------------------------------------------------------------------
140 --------------------------------------------------------------------}
141 -- | /O(n+m)/. See 'difference'.
142 (\\) :: IntSet -> IntSet -> IntSet
143 m1 \\ m2 = difference m1 m2
145 {--------------------------------------------------------------------
147 --------------------------------------------------------------------}
148 -- | A set of integers.
150 | Tip {-# UNPACK #-} !Int
151 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
156 {--------------------------------------------------------------------
158 --------------------------------------------------------------------}
159 -- | /O(1)/. Is the set empty?
160 null :: IntSet -> Bool
164 -- | /O(n)/. Cardinality of the set.
165 size :: IntSet -> Int
168 Bin p m l r -> size l + size r
172 -- | /O(min(n,W))/. Is the value a member of the set?
173 member :: Int -> IntSet -> Bool
177 | nomatch x p m -> False
178 | zero x m -> member x l
179 | otherwise -> member x r
183 -- 'lookup' is used by 'intersection' for left-biasing
184 lookup :: Int -> IntSet -> Maybe Int
186 = let nk = natFromInt k in seq nk (lookupN nk t)
188 lookupN :: Nat -> IntSet -> Maybe Int
192 | zeroN k (natFromInt m) -> lookupN k l
193 | otherwise -> lookupN k r
195 | (k == natFromInt kx) -> Just kx
196 | otherwise -> Nothing
199 {--------------------------------------------------------------------
201 --------------------------------------------------------------------}
202 -- | /O(1)/. The empty set.
207 -- | /O(1)/. A set of one element.
208 singleton :: Int -> IntSet
212 {--------------------------------------------------------------------
214 --------------------------------------------------------------------}
215 -- | /O(min(n,W))/. Add a value to the set. When the value is already
216 -- an element of the set, it is replaced by the new one, ie. 'insert'
218 insert :: Int -> IntSet -> IntSet
222 | nomatch x p m -> join x (Tip x) p t
223 | zero x m -> Bin p m (insert x l) r
224 | otherwise -> Bin p m l (insert x r)
227 | otherwise -> join x (Tip x) y t
230 -- right-biased insertion, used by 'union'
231 insertR :: Int -> IntSet -> IntSet
235 | nomatch x p m -> join x (Tip x) p t
236 | zero x m -> Bin p m (insert x l) r
237 | otherwise -> Bin p m l (insert x r)
240 | otherwise -> join x (Tip x) y t
243 -- | /O(min(n,W))/. Delete a value in the set. Returns the
244 -- original set when the value was not present.
245 delete :: Int -> IntSet -> IntSet
250 | zero x m -> bin p m (delete x l) r
251 | otherwise -> bin p m l (delete x r)
258 {--------------------------------------------------------------------
260 --------------------------------------------------------------------}
261 -- | The union of a list of sets.
262 unions :: [IntSet] -> IntSet
264 = foldlStrict union empty xs
267 -- | /O(n+m)/. The union of two sets.
268 union :: IntSet -> IntSet -> IntSet
269 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
270 | shorter m1 m2 = union1
271 | shorter m2 m1 = union2
272 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
273 | otherwise = join p1 t1 p2 t2
275 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
276 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
277 | otherwise = Bin p1 m1 l1 (union r1 t2)
279 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
280 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
281 | otherwise = Bin p2 m2 l2 (union t1 r2)
283 union (Tip x) t = insert x t
284 union t (Tip x) = insertR x t -- right bias
289 {--------------------------------------------------------------------
291 --------------------------------------------------------------------}
292 -- | /O(n+m)/. Difference between two sets.
293 difference :: IntSet -> IntSet -> IntSet
294 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
295 | shorter m1 m2 = difference1
296 | shorter m2 m1 = difference2
297 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
300 difference1 | nomatch p2 p1 m1 = t1
301 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
302 | otherwise = bin p1 m1 l1 (difference r1 t2)
304 difference2 | nomatch p1 p2 m2 = t1
305 | zero p1 m2 = difference t1 l2
306 | otherwise = difference t1 r2
308 difference t1@(Tip x) t2
312 difference Nil t = Nil
313 difference t (Tip x) = delete x t
318 {--------------------------------------------------------------------
320 --------------------------------------------------------------------}
321 -- | /O(n+m)/. The intersection of two sets.
322 intersection :: IntSet -> IntSet -> IntSet
323 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
324 | shorter m1 m2 = intersection1
325 | shorter m2 m1 = intersection2
326 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
329 intersection1 | nomatch p2 p1 m1 = Nil
330 | zero p2 m1 = intersection l1 t2
331 | otherwise = intersection r1 t2
333 intersection2 | nomatch p1 p2 m2 = Nil
334 | zero p1 m2 = intersection t1 l2
335 | otherwise = intersection t1 r2
337 intersection t1@(Tip x) t2
340 intersection t (Tip x)
344 intersection Nil t = Nil
345 intersection t Nil = Nil
349 {--------------------------------------------------------------------
351 --------------------------------------------------------------------}
352 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
353 isProperSubsetOf :: IntSet -> IntSet -> Bool
354 isProperSubsetOf t1 t2
355 = case subsetCmp t1 t2 of
359 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
361 | shorter m2 m1 = subsetCmpLt
362 | p1 == p2 = subsetCmpEq
363 | otherwise = GT -- disjoint
365 subsetCmpLt | nomatch p1 p2 m2 = GT
366 | zero p1 m2 = subsetCmp t1 l2
367 | otherwise = subsetCmp t1 r2
368 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
374 subsetCmp (Bin p m l r) t = GT
375 subsetCmp (Tip x) (Tip y)
377 | otherwise = GT -- disjoint
380 | otherwise = GT -- disjoint
381 subsetCmp Nil Nil = EQ
384 -- | /O(n+m)/. Is this a subset?
385 -- @(s1 `isSubsetOf` s2)@ tells whether s1 is a subset of s2.
387 isSubsetOf :: IntSet -> IntSet -> Bool
388 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
389 | shorter m1 m2 = False
390 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
391 else isSubsetOf t1 r2)
392 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
393 isSubsetOf (Bin p m l r) t = False
394 isSubsetOf (Tip x) t = member x t
395 isSubsetOf Nil t = True
398 {--------------------------------------------------------------------
400 --------------------------------------------------------------------}
401 -- | /O(n)/. Filter all elements that satisfy some predicate.
402 filter :: (Int -> Bool) -> IntSet -> IntSet
406 -> bin p m (filter pred l) (filter pred r)
412 -- | /O(n)/. partition the set according to some predicate.
413 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
417 -> let (l1,l2) = partition pred l
418 (r1,r2) = partition pred r
419 in (bin p m l1 r1, bin p m l2 r2)
422 | otherwise -> (Nil,t)
426 -- | /O(log n)/. The expression (@split x set@) is a pair @(set1,set2)@
427 -- where all elements in @set1@ are lower than @x@ and all elements in
428 -- @set2@ larger than @x@.
430 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
431 split :: Int -> IntSet -> (IntSet,IntSet)
435 | zero x m -> let (lt,gt) = split x l in (lt,union gt r)
436 | otherwise -> let (lt,gt) = split x r in (union l lt,gt)
440 | otherwise -> (Nil,Nil)
443 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
444 -- element was found in the original set.
445 splitMember :: Int -> IntSet -> (Bool,IntSet,IntSet)
449 | zero x m -> let (found,lt,gt) = splitMember x l in (found,lt,union gt r)
450 | otherwise -> let (found,lt,gt) = splitMember x r in (found,union l lt,gt)
452 | x>y -> (False,t,Nil)
453 | x<y -> (False,Nil,t)
454 | otherwise -> (True,Nil,Nil)
455 Nil -> (False,Nil,Nil)
457 {----------------------------------------------------------------------
459 ----------------------------------------------------------------------}
461 -- | /O(n*min(n,W))/.
462 -- @map f s@ is the set obtained by applying @f@ to each element of @s@.
464 -- It's worth noting that the size of the result may be smaller if,
465 -- for some @(x,y)@, @x \/= y && f x == f y@
467 map :: (Int->Int) -> IntSet -> IntSet
468 map f = fromList . List.map f . toList
470 {--------------------------------------------------------------------
472 --------------------------------------------------------------------}
473 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
475 -- > sum set == fold (+) 0 set
476 -- > elems set == fold (:) [] set
477 fold :: (Int -> b -> b) -> b -> IntSet -> b
481 foldr :: (Int -> b -> b) -> b -> IntSet -> b
484 Bin p m l r -> foldr f (foldr f z r) l
488 {--------------------------------------------------------------------
490 --------------------------------------------------------------------}
491 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
492 elems :: IntSet -> [Int]
496 {--------------------------------------------------------------------
498 --------------------------------------------------------------------}
499 -- | /O(n)/. Convert the set to a list of elements.
500 toList :: IntSet -> [Int]
504 -- | /O(n)/. Convert the set to an ascending list of elements.
505 toAscList :: IntSet -> [Int]
507 = -- NOTE: the following algorithm only works for big-endian trees
508 let (pos,neg) = span (>=0) (foldr (:) [] t) in neg ++ pos
510 -- | /O(n*min(n,W))/. Create a set from a list of integers.
511 fromList :: [Int] -> IntSet
513 = foldlStrict ins empty xs
517 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
518 fromAscList :: [Int] -> IntSet
522 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
523 fromDistinctAscList :: [Int] -> IntSet
524 fromDistinctAscList xs
528 {--------------------------------------------------------------------
530 --------------------------------------------------------------------}
531 instance Eq IntSet where
532 t1 == t2 = equal t1 t2
533 t1 /= t2 = nequal t1 t2
535 equal :: IntSet -> IntSet -> Bool
536 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
537 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
538 equal (Tip x) (Tip y)
543 nequal :: IntSet -> IntSet -> Bool
544 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
545 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
546 nequal (Tip x) (Tip y)
548 nequal Nil Nil = False
551 {--------------------------------------------------------------------
553 --------------------------------------------------------------------}
555 instance Ord IntSet where
556 compare s1 s2 = compare (toAscList s1) (toAscList s2)
557 -- tentative implementation. See if more efficient exists.
559 {--------------------------------------------------------------------
561 --------------------------------------------------------------------}
563 instance Monoid IntSet where
568 {--------------------------------------------------------------------
570 --------------------------------------------------------------------}
571 instance Show IntSet where
572 showsPrec d s = showSet (toList s)
574 showSet :: [Int] -> ShowS
578 = showChar '{' . shows x . showTail xs
580 showTail [] = showChar '}'
581 showTail (x:xs) = showChar ',' . shows x . showTail xs
583 {--------------------------------------------------------------------
585 --------------------------------------------------------------------}
586 -- | /O(n)/. Show the tree that implements the set. The tree is shown
587 -- in a compressed, hanging format.
588 showTree :: IntSet -> String
590 = showTreeWith True False s
593 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
594 the tree that implements the set. If @hang@ is
595 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
596 @wide@ is true, an extra wide version is shown.
598 showTreeWith :: Bool -> Bool -> IntSet -> String
599 showTreeWith hang wide t
600 | hang = (showsTreeHang wide [] t) ""
601 | otherwise = (showsTree wide [] [] t) ""
603 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
604 showsTree wide lbars rbars t
607 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
608 showWide wide rbars .
609 showsBars lbars . showString (showBin p m) . showString "\n" .
610 showWide wide lbars .
611 showsTree wide (withEmpty lbars) (withBar lbars) l
613 -> showsBars lbars . showString " " . shows x . showString "\n"
614 Nil -> showsBars lbars . showString "|\n"
616 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
617 showsTreeHang wide bars t
620 -> showsBars bars . showString (showBin p m) . showString "\n" .
622 showsTreeHang wide (withBar bars) l .
624 showsTreeHang wide (withEmpty bars) r
626 -> showsBars bars . showString " " . shows x . showString "\n"
627 Nil -> showsBars bars . showString "|\n"
630 = "*" -- ++ show (p,m)
633 | wide = showString (concat (reverse bars)) . showString "|\n"
636 showsBars :: [String] -> ShowS
640 _ -> showString (concat (reverse (tail bars))) . showString node
643 withBar bars = "| ":bars
644 withEmpty bars = " ":bars
647 {--------------------------------------------------------------------
649 --------------------------------------------------------------------}
650 {--------------------------------------------------------------------
652 --------------------------------------------------------------------}
653 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
655 | zero p1 m = Bin p m t1 t2
656 | otherwise = Bin p m t2 t1
661 {--------------------------------------------------------------------
662 @bin@ assures that we never have empty trees within a tree.
663 --------------------------------------------------------------------}
664 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
667 bin p m l r = Bin p m l r
670 {--------------------------------------------------------------------
671 Endian independent bit twiddling
672 --------------------------------------------------------------------}
673 zero :: Int -> Mask -> Bool
675 = (natFromInt i) .&. (natFromInt m) == 0
677 nomatch,match :: Int -> Prefix -> Mask -> Bool
684 mask :: Int -> Mask -> Prefix
686 = maskW (natFromInt i) (natFromInt m)
688 zeroN :: Nat -> Nat -> Bool
689 zeroN i m = (i .&. m) == 0
691 {--------------------------------------------------------------------
692 Big endian operations
693 --------------------------------------------------------------------}
694 maskW :: Nat -> Nat -> Prefix
696 = intFromNat (i .&. (complement (m-1) `xor` m))
698 shorter :: Mask -> Mask -> Bool
700 = (natFromInt m1) > (natFromInt m2)
702 branchMask :: Prefix -> Prefix -> Mask
704 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
706 {----------------------------------------------------------------------
707 Finding the highest bit (mask) in a word [x] can be done efficiently in
709 * convert to a floating point value and the mantissa tells us the
710 [log2(x)] that corresponds with the highest bit position. The mantissa
711 is retrieved either via the standard C function [frexp] or by some bit
712 twiddling on IEEE compatible numbers (float). Note that one needs to
713 use at least [double] precision for an accurate mantissa of 32 bit
715 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
716 * use processor specific assembler instruction (asm).
718 The most portable way would be [bit], but is it efficient enough?
719 I have measured the cycle counts of the different methods on an AMD
720 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
722 highestBitMask: method cycles
729 highestBit: method cycles
736 Wow, the bit twiddling is on today's RISC like machines even faster
737 than a single CISC instruction (BSR)!
738 ----------------------------------------------------------------------}
740 {----------------------------------------------------------------------
741 [highestBitMask] returns a word where only the highest bit is set.
742 It is found by first setting all bits in lower positions than the
743 highest bit and than taking an exclusive or with the original value.
744 Allthough the function may look expensive, GHC compiles this into
745 excellent C code that subsequently compiled into highly efficient
746 machine code. The algorithm is derived from Jorg Arndt's FXT library.
747 ----------------------------------------------------------------------}
748 highestBitMask :: Nat -> Nat
750 = case (x .|. shiftRL x 1) of
751 x -> case (x .|. shiftRL x 2) of
752 x -> case (x .|. shiftRL x 4) of
753 x -> case (x .|. shiftRL x 8) of
754 x -> case (x .|. shiftRL x 16) of
755 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
756 x -> (x `xor` (shiftRL x 1))
759 {--------------------------------------------------------------------
761 --------------------------------------------------------------------}
765 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
769 {--------------------------------------------------------------------
771 --------------------------------------------------------------------}
772 testTree :: [Int] -> IntSet
773 testTree xs = fromList xs
774 test1 = testTree [1..20]
775 test2 = testTree [30,29..10]
776 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
778 {--------------------------------------------------------------------
780 --------------------------------------------------------------------}
785 { configMaxTest = 500
786 , configMaxFail = 5000
787 , configSize = \n -> (div n 2 + 3)
788 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
792 {--------------------------------------------------------------------
793 Arbitrary, reasonably balanced trees
794 --------------------------------------------------------------------}
795 instance Arbitrary IntSet where
796 arbitrary = do{ xs <- arbitrary
797 ; return (fromList xs)
801 {--------------------------------------------------------------------
802 Single, Insert, Delete
803 --------------------------------------------------------------------}
804 prop_Single :: Int -> Bool
806 = (insert x empty == singleton x)
808 prop_InsertDelete :: Int -> IntSet -> Property
809 prop_InsertDelete k t
810 = not (member k t) ==> delete k (insert k t) == t
813 {--------------------------------------------------------------------
815 --------------------------------------------------------------------}
816 prop_UnionInsert :: Int -> IntSet -> Bool
818 = union t (singleton x) == insert x t
820 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
821 prop_UnionAssoc t1 t2 t3
822 = union t1 (union t2 t3) == union (union t1 t2) t3
824 prop_UnionComm :: IntSet -> IntSet -> Bool
826 = (union t1 t2 == union t2 t1)
828 prop_Diff :: [Int] -> [Int] -> Bool
830 = toAscList (difference (fromList xs) (fromList ys))
831 == List.sort ((List.\\) (nub xs) (nub ys))
833 prop_Int :: [Int] -> [Int] -> Bool
835 = toAscList (intersection (fromList xs) (fromList ys))
836 == List.sort (nub ((List.intersect) (xs) (ys)))
838 {--------------------------------------------------------------------
840 --------------------------------------------------------------------}
842 = forAll (choose (5,100)) $ \n ->
844 in fromAscList xs == fromList xs
846 prop_List :: [Int] -> Bool
848 = (sort (nub xs) == toAscList (fromList xs))