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 -}
119 {--------------------------------------------------------------------
121 * Older Hugs doesn't define 'Word'.
122 * Newer Hugs defines 'Word' in the Prelude but no operations.
123 --------------------------------------------------------------------}
124 type Nat = Word32 -- illegal on 64-bit platforms!
126 {--------------------------------------------------------------------
128 * A "Nat" is a natural machine word (an unsigned Int)
129 --------------------------------------------------------------------}
133 natFromInt :: Int -> Nat
134 natFromInt i = fromIntegral i
136 intFromNat :: Nat -> Int
137 intFromNat w = fromIntegral w
139 shiftRL :: Nat -> Int -> Nat
140 #if __GLASGOW_HASKELL__
141 {--------------------------------------------------------------------
142 GHC: use unboxing to get @shiftRL@ inlined.
143 --------------------------------------------------------------------}
144 shiftRL (W# x) (I# i)
147 shiftRL x i = shiftR x i
150 {--------------------------------------------------------------------
152 --------------------------------------------------------------------}
153 -- | /O(n+m)/. See 'difference'.
154 (\\) :: IntSet -> IntSet -> IntSet
155 m1 \\ m2 = difference m1 m2
157 {--------------------------------------------------------------------
159 --------------------------------------------------------------------}
160 -- | A set of integers.
162 | Tip {-# UNPACK #-} !Int
163 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
168 {--------------------------------------------------------------------
170 --------------------------------------------------------------------}
171 -- | /O(1)/. Is the set empty?
172 null :: IntSet -> Bool
176 -- | /O(n)/. Cardinality of the set.
177 size :: IntSet -> Int
180 Bin p m l r -> size l + size r
184 -- | /O(min(n,W))/. Is the value a member of the set?
185 member :: Int -> IntSet -> Bool
189 | nomatch x p m -> False
190 | zero x m -> member x l
191 | otherwise -> member x r
195 -- 'lookup' is used by 'intersection' for left-biasing
196 lookup :: Int -> IntSet -> Maybe Int
198 = let nk = natFromInt k in seq nk (lookupN nk t)
200 lookupN :: Nat -> IntSet -> Maybe Int
204 | zeroN k (natFromInt m) -> lookupN k l
205 | otherwise -> lookupN k r
207 | (k == natFromInt kx) -> Just kx
208 | otherwise -> Nothing
211 {--------------------------------------------------------------------
213 --------------------------------------------------------------------}
214 -- | /O(1)/. The empty set.
219 -- | /O(1)/. A set of one element.
220 singleton :: Int -> IntSet
224 {--------------------------------------------------------------------
226 --------------------------------------------------------------------}
227 -- | /O(min(n,W))/. Add a value to the set. When the value is already
228 -- an element of the set, it is replaced by the new one, ie. 'insert'
230 insert :: Int -> IntSet -> IntSet
234 | nomatch x p m -> join x (Tip x) p t
235 | zero x m -> Bin p m (insert x l) r
236 | otherwise -> Bin p m l (insert x r)
239 | otherwise -> join x (Tip x) y t
242 -- right-biased insertion, used by 'union'
243 insertR :: Int -> IntSet -> IntSet
247 | nomatch x p m -> join x (Tip x) p t
248 | zero x m -> Bin p m (insert x l) r
249 | otherwise -> Bin p m l (insert x r)
252 | otherwise -> join x (Tip x) y t
255 -- | /O(min(n,W))/. Delete a value in the set. Returns the
256 -- original set when the value was not present.
257 delete :: Int -> IntSet -> IntSet
262 | zero x m -> bin p m (delete x l) r
263 | otherwise -> bin p m l (delete x r)
270 {--------------------------------------------------------------------
272 --------------------------------------------------------------------}
273 -- | The union of a list of sets.
274 unions :: [IntSet] -> IntSet
276 = foldlStrict union empty xs
279 -- | /O(n+m)/. The union of two sets.
280 union :: IntSet -> IntSet -> IntSet
281 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
282 | shorter m1 m2 = union1
283 | shorter m2 m1 = union2
284 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
285 | otherwise = join p1 t1 p2 t2
287 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
288 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
289 | otherwise = Bin p1 m1 l1 (union r1 t2)
291 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
292 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
293 | otherwise = Bin p2 m2 l2 (union t1 r2)
295 union (Tip x) t = insert x t
296 union t (Tip x) = insertR x t -- right bias
301 {--------------------------------------------------------------------
303 --------------------------------------------------------------------}
304 -- | /O(n+m)/. Difference between two sets.
305 difference :: IntSet -> IntSet -> IntSet
306 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
307 | shorter m1 m2 = difference1
308 | shorter m2 m1 = difference2
309 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
312 difference1 | nomatch p2 p1 m1 = t1
313 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
314 | otherwise = bin p1 m1 l1 (difference r1 t2)
316 difference2 | nomatch p1 p2 m2 = t1
317 | zero p1 m2 = difference t1 l2
318 | otherwise = difference t1 r2
320 difference t1@(Tip x) t2
324 difference Nil t = Nil
325 difference t (Tip x) = delete x t
330 {--------------------------------------------------------------------
332 --------------------------------------------------------------------}
333 -- | /O(n+m)/. The intersection of two sets.
334 intersection :: IntSet -> IntSet -> IntSet
335 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
336 | shorter m1 m2 = intersection1
337 | shorter m2 m1 = intersection2
338 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
341 intersection1 | nomatch p2 p1 m1 = Nil
342 | zero p2 m1 = intersection l1 t2
343 | otherwise = intersection r1 t2
345 intersection2 | nomatch p1 p2 m2 = Nil
346 | zero p1 m2 = intersection t1 l2
347 | otherwise = intersection t1 r2
349 intersection t1@(Tip x) t2
352 intersection t (Tip x)
356 intersection Nil t = Nil
357 intersection t Nil = Nil
361 {--------------------------------------------------------------------
363 --------------------------------------------------------------------}
364 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
365 isProperSubsetOf :: IntSet -> IntSet -> Bool
366 isProperSubsetOf t1 t2
367 = case subsetCmp t1 t2 of
371 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
373 | shorter m2 m1 = subsetCmpLt
374 | p1 == p2 = subsetCmpEq
375 | otherwise = GT -- disjoint
377 subsetCmpLt | nomatch p1 p2 m2 = GT
378 | zero p1 m2 = subsetCmp t1 l2
379 | otherwise = subsetCmp t1 r2
380 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
386 subsetCmp (Bin p m l r) t = GT
387 subsetCmp (Tip x) (Tip y)
389 | otherwise = GT -- disjoint
392 | otherwise = GT -- disjoint
393 subsetCmp Nil Nil = EQ
396 -- | /O(n+m)/. Is this a subset?
397 -- @(s1 `isSubsetOf` s2)@ tells whether s1 is a subset of s2.
399 isSubsetOf :: IntSet -> IntSet -> Bool
400 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
401 | shorter m1 m2 = False
402 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
403 else isSubsetOf t1 r2)
404 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
405 isSubsetOf (Bin p m l r) t = False
406 isSubsetOf (Tip x) t = member x t
407 isSubsetOf Nil t = True
410 {--------------------------------------------------------------------
412 --------------------------------------------------------------------}
413 -- | /O(n)/. Filter all elements that satisfy some predicate.
414 filter :: (Int -> Bool) -> IntSet -> IntSet
418 -> bin p m (filter pred l) (filter pred r)
424 -- | /O(n)/. partition the set according to some predicate.
425 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
429 -> let (l1,l2) = partition pred l
430 (r1,r2) = partition pred r
431 in (bin p m l1 r1, bin p m l2 r2)
434 | otherwise -> (Nil,t)
438 -- | /O(log n)/. The expression (@split x set@) is a pair @(set1,set2)@
439 -- where all elements in @set1@ are lower than @x@ and all elements in
440 -- @set2@ larger than @x@.
442 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
443 split :: Int -> IntSet -> (IntSet,IntSet)
447 | zero x m -> let (lt,gt) = split x l in (lt,union gt r)
448 | otherwise -> let (lt,gt) = split x r in (union l lt,gt)
452 | otherwise -> (Nil,Nil)
455 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
456 -- element was found in the original set.
457 splitMember :: Int -> IntSet -> (Bool,IntSet,IntSet)
461 | zero x m -> let (found,lt,gt) = splitMember x l in (found,lt,union gt r)
462 | otherwise -> let (found,lt,gt) = splitMember x r in (found,union l lt,gt)
464 | x>y -> (False,t,Nil)
465 | x<y -> (False,Nil,t)
466 | otherwise -> (True,Nil,Nil)
467 Nil -> (False,Nil,Nil)
469 {----------------------------------------------------------------------
471 ----------------------------------------------------------------------}
473 -- | /O(n*min(n,W))/.
474 -- @map f s@ is the set obtained by applying @f@ to each element of @s@.
476 -- It's worth noting that the size of the result may be smaller if,
477 -- for some @(x,y)@, @x \/= y && f x == f y@
479 map :: (Int->Int) -> IntSet -> IntSet
480 map f = fromList . List.map f . toList
482 {--------------------------------------------------------------------
484 --------------------------------------------------------------------}
485 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
487 -- > sum set == fold (+) 0 set
488 -- > elems set == fold (:) [] set
489 fold :: (Int -> b -> b) -> b -> IntSet -> b
493 foldr :: (Int -> b -> b) -> b -> IntSet -> b
496 Bin p m l r -> foldr f (foldr f z r) l
500 {--------------------------------------------------------------------
502 --------------------------------------------------------------------}
503 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
504 elems :: IntSet -> [Int]
508 {--------------------------------------------------------------------
510 --------------------------------------------------------------------}
511 -- | /O(n)/. Convert the set to a list of elements.
512 toList :: IntSet -> [Int]
516 -- | /O(n)/. Convert the set to an ascending list of elements.
517 toAscList :: IntSet -> [Int]
519 = -- NOTE: the following algorithm only works for big-endian trees
520 let (pos,neg) = span (>=0) (foldr (:) [] t) in neg ++ pos
522 -- | /O(n*min(n,W))/. Create a set from a list of integers.
523 fromList :: [Int] -> IntSet
525 = foldlStrict ins empty xs
529 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
530 fromAscList :: [Int] -> IntSet
534 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
535 fromDistinctAscList :: [Int] -> IntSet
536 fromDistinctAscList xs
540 {--------------------------------------------------------------------
542 --------------------------------------------------------------------}
543 instance Eq IntSet where
544 t1 == t2 = equal t1 t2
545 t1 /= t2 = nequal t1 t2
547 equal :: IntSet -> IntSet -> Bool
548 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
549 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
550 equal (Tip x) (Tip y)
555 nequal :: IntSet -> IntSet -> Bool
556 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
557 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
558 nequal (Tip x) (Tip y)
560 nequal Nil Nil = False
563 {--------------------------------------------------------------------
565 --------------------------------------------------------------------}
567 instance Ord IntSet where
568 compare s1 s2 = compare (toAscList s1) (toAscList s2)
569 -- tentative implementation. See if more efficient exists.
571 {--------------------------------------------------------------------
573 --------------------------------------------------------------------}
575 instance Monoid IntSet where
580 {--------------------------------------------------------------------
582 --------------------------------------------------------------------}
583 instance Show IntSet where
584 showsPrec d s = showSet (toList s)
586 showSet :: [Int] -> ShowS
590 = showChar '{' . shows x . showTail xs
592 showTail [] = showChar '}'
593 showTail (x:xs) = showChar ',' . shows x . showTail xs
595 {--------------------------------------------------------------------
597 --------------------------------------------------------------------}
598 -- | /O(n)/. Show the tree that implements the set. The tree is shown
599 -- in a compressed, hanging format.
600 showTree :: IntSet -> String
602 = showTreeWith True False s
605 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
606 the tree that implements the set. If @hang@ is
607 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
608 @wide@ is true, an extra wide version is shown.
610 showTreeWith :: Bool -> Bool -> IntSet -> String
611 showTreeWith hang wide t
612 | hang = (showsTreeHang wide [] t) ""
613 | otherwise = (showsTree wide [] [] t) ""
615 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
616 showsTree wide lbars rbars t
619 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
620 showWide wide rbars .
621 showsBars lbars . showString (showBin p m) . showString "\n" .
622 showWide wide lbars .
623 showsTree wide (withEmpty lbars) (withBar lbars) l
625 -> showsBars lbars . showString " " . shows x . showString "\n"
626 Nil -> showsBars lbars . showString "|\n"
628 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
629 showsTreeHang wide bars t
632 -> showsBars bars . showString (showBin p m) . showString "\n" .
634 showsTreeHang wide (withBar bars) l .
636 showsTreeHang wide (withEmpty bars) r
638 -> showsBars bars . showString " " . shows x . showString "\n"
639 Nil -> showsBars bars . showString "|\n"
642 = "*" -- ++ show (p,m)
645 | wide = showString (concat (reverse bars)) . showString "|\n"
648 showsBars :: [String] -> ShowS
652 _ -> showString (concat (reverse (tail bars))) . showString node
655 withBar bars = "| ":bars
656 withEmpty bars = " ":bars
659 {--------------------------------------------------------------------
661 --------------------------------------------------------------------}
662 {--------------------------------------------------------------------
664 --------------------------------------------------------------------}
665 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
667 | zero p1 m = Bin p m t1 t2
668 | otherwise = Bin p m t2 t1
673 {--------------------------------------------------------------------
674 @bin@ assures that we never have empty trees within a tree.
675 --------------------------------------------------------------------}
676 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
679 bin p m l r = Bin p m l r
682 {--------------------------------------------------------------------
683 Endian independent bit twiddling
684 --------------------------------------------------------------------}
685 zero :: Int -> Mask -> Bool
687 = (natFromInt i) .&. (natFromInt m) == 0
689 nomatch,match :: Int -> Prefix -> Mask -> Bool
696 mask :: Int -> Mask -> Prefix
698 = maskW (natFromInt i) (natFromInt m)
700 zeroN :: Nat -> Nat -> Bool
701 zeroN i m = (i .&. m) == 0
703 {--------------------------------------------------------------------
704 Big endian operations
705 --------------------------------------------------------------------}
706 maskW :: Nat -> Nat -> Prefix
708 = intFromNat (i .&. (complement (m-1) `xor` m))
710 shorter :: Mask -> Mask -> Bool
712 = (natFromInt m1) > (natFromInt m2)
714 branchMask :: Prefix -> Prefix -> Mask
716 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
718 {----------------------------------------------------------------------
719 Finding the highest bit (mask) in a word [x] can be done efficiently in
721 * convert to a floating point value and the mantissa tells us the
722 [log2(x)] that corresponds with the highest bit position. The mantissa
723 is retrieved either via the standard C function [frexp] or by some bit
724 twiddling on IEEE compatible numbers (float). Note that one needs to
725 use at least [double] precision for an accurate mantissa of 32 bit
727 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
728 * use processor specific assembler instruction (asm).
730 The most portable way would be [bit], but is it efficient enough?
731 I have measured the cycle counts of the different methods on an AMD
732 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
734 highestBitMask: method cycles
741 highestBit: method cycles
748 Wow, the bit twiddling is on today's RISC like machines even faster
749 than a single CISC instruction (BSR)!
750 ----------------------------------------------------------------------}
752 {----------------------------------------------------------------------
753 [highestBitMask] returns a word where only the highest bit is set.
754 It is found by first setting all bits in lower positions than the
755 highest bit and than taking an exclusive or with the original value.
756 Allthough the function may look expensive, GHC compiles this into
757 excellent C code that subsequently compiled into highly efficient
758 machine code. The algorithm is derived from Jorg Arndt's FXT library.
759 ----------------------------------------------------------------------}
760 highestBitMask :: Nat -> Nat
762 = case (x .|. shiftRL x 1) of
763 x -> case (x .|. shiftRL x 2) of
764 x -> case (x .|. shiftRL x 4) of
765 x -> case (x .|. shiftRL x 8) of
766 x -> case (x .|. shiftRL x 16) of
767 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
768 x -> (x `xor` (shiftRL x 1))
771 {--------------------------------------------------------------------
773 --------------------------------------------------------------------}
777 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
781 {--------------------------------------------------------------------
783 --------------------------------------------------------------------}
784 testTree :: [Int] -> IntSet
785 testTree xs = fromList xs
786 test1 = testTree [1..20]
787 test2 = testTree [30,29..10]
788 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
790 {--------------------------------------------------------------------
792 --------------------------------------------------------------------}
797 { configMaxTest = 500
798 , configMaxFail = 5000
799 , configSize = \n -> (div n 2 + 3)
800 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
804 {--------------------------------------------------------------------
805 Arbitrary, reasonably balanced trees
806 --------------------------------------------------------------------}
807 instance Arbitrary IntSet where
808 arbitrary = do{ xs <- arbitrary
809 ; return (fromList xs)
813 {--------------------------------------------------------------------
814 Single, Insert, Delete
815 --------------------------------------------------------------------}
816 prop_Single :: Int -> Bool
818 = (insert x empty == singleton x)
820 prop_InsertDelete :: Int -> IntSet -> Property
821 prop_InsertDelete k t
822 = not (member k t) ==> delete k (insert k t) == t
825 {--------------------------------------------------------------------
827 --------------------------------------------------------------------}
828 prop_UnionInsert :: Int -> IntSet -> Bool
830 = union t (singleton x) == insert x t
832 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
833 prop_UnionAssoc t1 t2 t3
834 = union t1 (union t2 t3) == union (union t1 t2) t3
836 prop_UnionComm :: IntSet -> IntSet -> Bool
838 = (union t1 t2 == union t2 t1)
840 prop_Diff :: [Int] -> [Int] -> Bool
842 = toAscList (difference (fromList xs) (fromList ys))
843 == List.sort ((List.\\) (nub xs) (nub ys))
845 prop_Int :: [Int] -> [Int] -> Bool
847 = toAscList (intersection (fromList xs) (fromList ys))
848 == List.sort (nub ((List.intersect) (xs) (ys)))
850 {--------------------------------------------------------------------
852 --------------------------------------------------------------------}
854 = forAll (choose (5,100)) $ \n ->
856 in fromAscList xs == fromList xs
858 prop_List :: [Int] -> Bool
860 = (sort (nub xs) == toAscList (fromList xs))