1 -----------------------------------------------------------------------------
4 -- Copyright : (c) Daan Leijen 2002
6 -- Maintainer : libraries@haskell.org
7 -- Stability : provisional
8 -- Portability : portable
10 -- An efficient implementation of sets.
12 -- Since many function names (but not the type name) clash with
13 -- "Prelude" names, this module is usually imported @qualified@, e.g.
15 -- > import Data.Set (Set)
16 -- > import qualified Data.Set as Set
18 -- The implementation of 'Set' is based on /size balanced/ binary trees (or
19 -- trees of /bounded balance/) as described by:
21 -- * Stephen Adams, \"/Efficient sets: a balancing act/\",
22 -- Journal of Functional Programming 3(4):553-562, October 1993,
23 -- <http://www.swiss.ai.mit.edu/~adams/BB>.
25 -- * J. Nievergelt and E.M. Reingold,
26 -- \"/Binary search trees of bounded balance/\",
27 -- SIAM journal of computing 2(1), March 1973.
29 -- Note that the implementation is /left-biased/ -- the elements of a
30 -- first argument are always preferred to the second, for example in
31 -- 'union' or 'insert'. Of course, left-biasing can only be observed
32 -- when equality is an equivalence relation instead of structural
34 -----------------------------------------------------------------------------
38 Set -- instance Eq,Ord,Show,Read,Data,Typeable
103 import Prelude hiding (filter,foldr,null,map)
104 import qualified Data.List as List
105 import Data.Monoid (Monoid(..))
107 import Data.Foldable (Foldable(foldMap))
112 import List (nub,sort)
113 import qualified List
116 #if __GLASGOW_HASKELL__
118 import Data.Generics.Basics
119 import Data.Generics.Instances
122 {--------------------------------------------------------------------
124 --------------------------------------------------------------------}
127 -- | /O(n+m)/. See 'difference'.
128 (\\) :: Ord a => Set a -> Set a -> Set a
129 m1 \\ m2 = difference m1 m2
131 {--------------------------------------------------------------------
132 Sets are size balanced trees
133 --------------------------------------------------------------------}
134 -- | A set of values @a@.
136 | Bin {-# UNPACK #-} !Size a !(Set a) !(Set a)
140 instance Ord a => Monoid (Set a) where
145 instance Foldable Set where
146 foldMap f Tip = mempty
147 foldMap f (Bin _s k l r) = foldMap f l `mappend` f k `mappend` foldMap f r
149 #if __GLASGOW_HASKELL__
151 {--------------------------------------------------------------------
153 --------------------------------------------------------------------}
155 -- This instance preserves data abstraction at the cost of inefficiency.
156 -- We omit reflection services for the sake of data abstraction.
158 instance (Data a, Ord a) => Data (Set a) where
159 gfoldl f z set = z fromList `f` (toList set)
160 toConstr _ = error "toConstr"
161 gunfold _ _ = error "gunfold"
162 dataTypeOf _ = mkNorepType "Data.Set.Set"
163 dataCast1 f = gcast1 f
167 {--------------------------------------------------------------------
169 --------------------------------------------------------------------}
170 -- | /O(1)/. Is this the empty set?
171 null :: Set a -> Bool
175 Bin sz x l r -> False
177 -- | /O(1)/. The number of elements in the set.
184 -- | /O(log n)/. Is the element in the set?
185 member :: Ord a => a -> Set a -> Bool
190 -> case compare x y of
195 -- | /O(log n)/. Is the element not in the set?
196 notMember :: Ord a => a -> Set a -> Bool
197 notMember x t = not $ member x t
199 {--------------------------------------------------------------------
201 --------------------------------------------------------------------}
202 -- | /O(1)/. The empty set.
207 -- | /O(1)/. Create a singleton set.
208 singleton :: a -> Set a
212 {--------------------------------------------------------------------
214 --------------------------------------------------------------------}
215 -- | /O(log n)/. Insert an element in a set.
216 -- If the set already contains an element equal to the given value,
217 -- it is replaced with the new value.
218 insert :: Ord a => a -> Set a -> Set a
223 -> case compare x y of
224 LT -> balance y (insert x l) r
225 GT -> balance y l (insert x r)
229 -- | /O(log n)/. Delete an element from a set.
230 delete :: Ord a => a -> Set a -> Set a
235 -> case compare x y of
236 LT -> balance y (delete x l) r
237 GT -> balance y l (delete x r)
240 {--------------------------------------------------------------------
242 --------------------------------------------------------------------}
243 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
244 isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
245 isProperSubsetOf s1 s2
246 = (size s1 < size s2) && (isSubsetOf s1 s2)
249 -- | /O(n+m)/. Is this a subset?
250 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
251 isSubsetOf :: Ord a => Set a -> Set a -> Bool
253 = (size t1 <= size t2) && (isSubsetOfX t1 t2)
255 isSubsetOfX Tip t = True
256 isSubsetOfX t Tip = False
257 isSubsetOfX (Bin _ x l r) t
258 = found && isSubsetOfX l lt && isSubsetOfX r gt
260 (lt,found,gt) = splitMember x t
263 {--------------------------------------------------------------------
265 --------------------------------------------------------------------}
266 -- | /O(log n)/. The minimal element of a set.
267 findMin :: Set a -> a
268 findMin (Bin _ x Tip r) = x
269 findMin (Bin _ x l r) = findMin l
270 findMin Tip = error "Set.findMin: empty set has no minimal element"
272 -- | /O(log n)/. The maximal element of a set.
273 findMax :: Set a -> a
274 findMax (Bin _ x l Tip) = x
275 findMax (Bin _ x l r) = findMax r
276 findMax Tip = error "Set.findMax: empty set has no maximal element"
278 -- | /O(log n)/. Delete the minimal element.
279 deleteMin :: Set a -> Set a
280 deleteMin (Bin _ x Tip r) = r
281 deleteMin (Bin _ x l r) = balance x (deleteMin l) r
284 -- | /O(log n)/. Delete the maximal element.
285 deleteMax :: Set a -> Set a
286 deleteMax (Bin _ x l Tip) = l
287 deleteMax (Bin _ x l r) = balance x l (deleteMax r)
291 {--------------------------------------------------------------------
293 --------------------------------------------------------------------}
294 -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@).
295 unions :: Ord a => [Set a] -> Set a
297 = foldlStrict union empty ts
300 -- | /O(n+m)/. The union of two sets, preferring the first set when
301 -- equal elements are encountered.
302 -- The implementation uses the efficient /hedge-union/ algorithm.
303 -- Hedge-union is more efficient on (bigset `union` smallset).
304 union :: Ord a => Set a -> Set a -> Set a
307 union t1 t2 = hedgeUnion (const LT) (const GT) t1 t2
309 hedgeUnion cmplo cmphi t1 Tip
311 hedgeUnion cmplo cmphi Tip (Bin _ x l r)
312 = join x (filterGt cmplo l) (filterLt cmphi r)
313 hedgeUnion cmplo cmphi (Bin _ x l r) t2
314 = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2))
315 (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2))
319 {--------------------------------------------------------------------
321 --------------------------------------------------------------------}
322 -- | /O(n+m)/. Difference of two sets.
323 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
324 difference :: Ord a => Set a -> Set a -> Set a
325 difference Tip t2 = Tip
326 difference t1 Tip = t1
327 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
329 hedgeDiff cmplo cmphi Tip t
331 hedgeDiff cmplo cmphi (Bin _ x l r) Tip
332 = join x (filterGt cmplo l) (filterLt cmphi r)
333 hedgeDiff cmplo cmphi t (Bin _ x l r)
334 = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l)
335 (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r)
339 {--------------------------------------------------------------------
341 --------------------------------------------------------------------}
342 -- | /O(n+m)/. The intersection of two sets.
343 -- Elements of the result come from the first set, so for example
345 -- > import qualified Data.Set as S
346 -- > data AB = A | B deriving Show
347 -- > instance Ord AB where compare _ _ = EQ
348 -- > instance Eq AB where _ == _ = True
349 -- > main = print (S.singleton A `S.intersection` S.singleton B,
350 -- > S.singleton B `S.intersection` S.singleton A)
352 -- prints @(fromList [A],fromList [B])@.
353 intersection :: Ord a => Set a -> Set a -> Set a
354 intersection Tip t = Tip
355 intersection t Tip = Tip
356 intersection t1@(Bin s1 x1 l1 r1) t2@(Bin s2 x2 l2 r2) =
358 let (lt,found,gt) = splitLookup x2 t1
359 tl = intersection lt l2
360 tr = intersection gt r2
362 Just x -> join x tl tr
363 Nothing -> merge tl tr
364 else let (lt,found,gt) = splitMember x1 t2
365 tl = intersection l1 lt
366 tr = intersection r1 gt
367 in if found then join x1 tl tr
370 {--------------------------------------------------------------------
372 --------------------------------------------------------------------}
373 -- | /O(n)/. Filter all elements that satisfy the predicate.
374 filter :: Ord a => (a -> Bool) -> Set a -> Set a
376 filter p (Bin _ x l r)
377 | p x = join x (filter p l) (filter p r)
378 | otherwise = merge (filter p l) (filter p r)
380 -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
381 -- the predicate and one with all elements that don't satisfy the predicate.
383 partition :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a)
384 partition p Tip = (Tip,Tip)
385 partition p (Bin _ x l r)
386 | p x = (join x l1 r1,merge l2 r2)
387 | otherwise = (merge l1 r1,join x l2 r2)
389 (l1,l2) = partition p l
390 (r1,r2) = partition p r
392 {----------------------------------------------------------------------
394 ----------------------------------------------------------------------}
397 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
399 -- It's worth noting that the size of the result may be smaller if,
400 -- for some @(x,y)@, @x \/= y && f x == f y@
402 map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b
403 map f = fromList . List.map f . toList
407 -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.
408 -- /The precondition is not checked./
409 -- Semi-formally, we have:
411 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
412 -- > ==> mapMonotonic f s == map f s
413 -- > where ls = toList s
415 mapMonotonic :: (a->b) -> Set a -> Set b
416 mapMonotonic f Tip = Tip
417 mapMonotonic f (Bin sz x l r) =
418 Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)
421 {--------------------------------------------------------------------
423 --------------------------------------------------------------------}
424 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
425 fold :: (a -> b -> b) -> b -> Set a -> b
429 -- | /O(n)/. Post-order fold.
430 foldr :: (a -> b -> b) -> b -> Set a -> b
432 foldr f z (Bin _ x l r) = foldr f (f x (foldr f z r)) l
434 {--------------------------------------------------------------------
436 --------------------------------------------------------------------}
437 -- | /O(n)/. The elements of a set.
438 elems :: Set a -> [a]
442 {--------------------------------------------------------------------
444 --------------------------------------------------------------------}
445 -- | /O(n)/. Convert the set to a list of elements.
446 toList :: Set a -> [a]
450 -- | /O(n)/. Convert the set to an ascending list of elements.
451 toAscList :: Set a -> [a]
456 -- | /O(n*log n)/. Create a set from a list of elements.
457 fromList :: Ord a => [a] -> Set a
459 = foldlStrict ins empty xs
463 {--------------------------------------------------------------------
464 Building trees from ascending/descending lists can be done in linear time.
466 Note that if [xs] is ascending that:
467 fromAscList xs == fromList xs
468 --------------------------------------------------------------------}
469 -- | /O(n)/. Build a set from an ascending list in linear time.
470 -- /The precondition (input list is ascending) is not checked./
471 fromAscList :: Eq a => [a] -> Set a
473 = fromDistinctAscList (combineEq xs)
475 -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
480 (x:xx) -> combineEq' x xx
482 combineEq' z [] = [z]
484 | z==x = combineEq' z xs
485 | otherwise = z:combineEq' x xs
488 -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
489 -- /The precondition (input list is strictly ascending) is not checked./
490 fromDistinctAscList :: [a] -> Set a
491 fromDistinctAscList xs
492 = build const (length xs) xs
494 -- 1) use continutations so that we use heap space instead of stack space.
495 -- 2) special case for n==5 to build bushier trees.
496 build c 0 xs = c Tip xs
497 build c 5 xs = case xs of
499 -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx
500 build c n xs = seq nr $ build (buildR nr c) nl xs
505 buildR n c l (x:ys) = build (buildB l x c) n ys
506 buildB l x c r zs = c (bin x l r) zs
508 {--------------------------------------------------------------------
509 Eq converts the set to a list. In a lazy setting, this
510 actually seems one of the faster methods to compare two trees
511 and it is certainly the simplest :-)
512 --------------------------------------------------------------------}
513 instance Eq a => Eq (Set a) where
514 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
516 {--------------------------------------------------------------------
518 --------------------------------------------------------------------}
520 instance Ord a => Ord (Set a) where
521 compare s1 s2 = compare (toAscList s1) (toAscList s2)
523 {--------------------------------------------------------------------
525 --------------------------------------------------------------------}
526 instance Show a => Show (Set a) where
527 showsPrec p xs = showParen (p > 10) $
528 showString "fromList " . shows (toList xs)
530 showSet :: (Show a) => [a] -> ShowS
534 = showChar '{' . shows x . showTail xs
536 showTail [] = showChar '}'
537 showTail (x:xs) = showChar ',' . shows x . showTail xs
539 {--------------------------------------------------------------------
541 --------------------------------------------------------------------}
542 instance (Read a, Ord a) => Read (Set a) where
543 #ifdef __GLASGOW_HASKELL__
544 readPrec = parens $ prec 10 $ do
545 Ident "fromList" <- lexP
549 readListPrec = readListPrecDefault
551 readsPrec p = readParen (p > 10) $ \ r -> do
552 ("fromList",s) <- lex r
554 return (fromList xs,t)
557 {--------------------------------------------------------------------
559 --------------------------------------------------------------------}
561 #include "Typeable.h"
562 INSTANCE_TYPEABLE1(Set,setTc,"Set")
564 {--------------------------------------------------------------------
565 Utility functions that return sub-ranges of the original
566 tree. Some functions take a comparison function as argument to
567 allow comparisons against infinite values. A function [cmplo x]
568 should be read as [compare lo x].
570 [trim cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
571 and [cmphi x == GT] for the value [x] of the root.
572 [filterGt cmp t] A tree where for all values [k]. [cmp k == LT]
573 [filterLt cmp t] A tree where for all values [k]. [cmp k == GT]
575 [split k t] Returns two trees [l] and [r] where all values
576 in [l] are <[k] and all keys in [r] are >[k].
577 [splitMember k t] Just like [split] but also returns whether [k]
578 was found in the tree.
579 --------------------------------------------------------------------}
581 {--------------------------------------------------------------------
582 [trim lo hi t] trims away all subtrees that surely contain no
583 values between the range [lo] to [hi]. The returned tree is either
584 empty or the key of the root is between @lo@ and @hi@.
585 --------------------------------------------------------------------}
586 trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a
587 trim cmplo cmphi Tip = Tip
588 trim cmplo cmphi t@(Bin sx x l r)
590 LT -> case cmphi x of
592 le -> trim cmplo cmphi l
593 ge -> trim cmplo cmphi r
595 trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a)
596 trimMemberLo lo cmphi Tip = (False,Tip)
597 trimMemberLo lo cmphi t@(Bin sx x l r)
598 = case compare lo x of
599 LT -> case cmphi x of
600 GT -> (member lo t, t)
601 le -> trimMemberLo lo cmphi l
602 GT -> trimMemberLo lo cmphi r
603 EQ -> (True,trim (compare lo) cmphi r)
606 {--------------------------------------------------------------------
607 [filterGt x t] filter all values >[x] from tree [t]
608 [filterLt x t] filter all values <[x] from tree [t]
609 --------------------------------------------------------------------}
610 filterGt :: (a -> Ordering) -> Set a -> Set a
611 filterGt cmp Tip = Tip
612 filterGt cmp (Bin sx x l r)
614 LT -> join x (filterGt cmp l) r
618 filterLt :: (a -> Ordering) -> Set a -> Set a
619 filterLt cmp Tip = Tip
620 filterLt cmp (Bin sx x l r)
623 GT -> join x l (filterLt cmp r)
627 {--------------------------------------------------------------------
629 --------------------------------------------------------------------}
630 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
631 -- where all elements in @set1@ are lower than @x@ and all elements in
632 -- @set2@ larger than @x@. @x@ is not found in neither @set1@ nor @set2@.
633 split :: Ord a => a -> Set a -> (Set a,Set a)
634 split x Tip = (Tip,Tip)
635 split x (Bin sy y l r)
636 = case compare x y of
637 LT -> let (lt,gt) = split x l in (lt,join y gt r)
638 GT -> let (lt,gt) = split x r in (join y l lt,gt)
641 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
642 -- element was found in the original set.
643 splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a)
644 splitMember x t = let (l,m,r) = splitLookup x t in
645 (l,maybe False (const True) m,r)
647 -- | /O(log n)/. Performs a 'split' but also returns the pivot
648 -- element that was found in the original set.
649 splitLookup :: Ord a => a -> Set a -> (Set a,Maybe a,Set a)
650 splitLookup x Tip = (Tip,Nothing,Tip)
651 splitLookup x (Bin sy y l r)
652 = case compare x y of
653 LT -> let (lt,found,gt) = splitLookup x l in (lt,found,join y gt r)
654 GT -> let (lt,found,gt) = splitLookup x r in (join y l lt,found,gt)
657 {--------------------------------------------------------------------
658 Utility functions that maintain the balance properties of the tree.
659 All constructors assume that all values in [l] < [x] and all values
660 in [r] > [x], and that [l] and [r] are valid trees.
662 In order of sophistication:
663 [Bin sz x l r] The type constructor.
664 [bin x l r] Maintains the correct size, assumes that both [l]
665 and [r] are balanced with respect to each other.
666 [balance x l r] Restores the balance and size.
667 Assumes that the original tree was balanced and
668 that [l] or [r] has changed by at most one element.
669 [join x l r] Restores balance and size.
671 Furthermore, we can construct a new tree from two trees. Both operations
672 assume that all values in [l] < all values in [r] and that [l] and [r]
674 [glue l r] Glues [l] and [r] together. Assumes that [l] and
675 [r] are already balanced with respect to each other.
676 [merge l r] Merges two trees and restores balance.
678 Note: in contrast to Adam's paper, we use (<=) comparisons instead
679 of (<) comparisons in [join], [merge] and [balance].
680 Quickcheck (on [difference]) showed that this was necessary in order
681 to maintain the invariants. It is quite unsatisfactory that I haven't
682 been able to find out why this is actually the case! Fortunately, it
683 doesn't hurt to be a bit more conservative.
684 --------------------------------------------------------------------}
686 {--------------------------------------------------------------------
688 --------------------------------------------------------------------}
689 join :: a -> Set a -> Set a -> Set a
690 join x Tip r = insertMin x r
691 join x l Tip = insertMax x l
692 join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
693 | delta*sizeL <= sizeR = balance z (join x l lz) rz
694 | delta*sizeR <= sizeL = balance y ly (join x ry r)
695 | otherwise = bin x l r
698 -- insertMin and insertMax don't perform potentially expensive comparisons.
699 insertMax,insertMin :: a -> Set a -> Set a
704 -> balance y l (insertMax x r)
710 -> balance y (insertMin x l) r
712 {--------------------------------------------------------------------
713 [merge l r]: merges two trees.
714 --------------------------------------------------------------------}
715 merge :: Set a -> Set a -> Set a
718 merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
719 | delta*sizeL <= sizeR = balance y (merge l ly) ry
720 | delta*sizeR <= sizeL = balance x lx (merge rx r)
721 | otherwise = glue l r
723 {--------------------------------------------------------------------
724 [glue l r]: glues two trees together.
725 Assumes that [l] and [r] are already balanced with respect to each other.
726 --------------------------------------------------------------------}
727 glue :: Set a -> Set a -> Set a
731 | size l > size r = let (m,l') = deleteFindMax l in balance m l' r
732 | otherwise = let (m,r') = deleteFindMin r in balance m l r'
735 -- | /O(log n)/. Delete and find the minimal element.
737 -- > deleteFindMin set = (findMin set, deleteMin set)
739 deleteFindMin :: Set a -> (a,Set a)
742 Bin _ x Tip r -> (x,r)
743 Bin _ x l r -> let (xm,l') = deleteFindMin l in (xm,balance x l' r)
744 Tip -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)
746 -- | /O(log n)/. Delete and find the maximal element.
748 -- > deleteFindMax set = (findMax set, deleteMax set)
749 deleteFindMax :: Set a -> (a,Set a)
752 Bin _ x l Tip -> (x,l)
753 Bin _ x l r -> let (xm,r') = deleteFindMax r in (xm,balance x l r')
754 Tip -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)
756 -- | /O(log n)/. Retrieves the minimal key of the set, and the set stripped from that element
757 -- @fail@s (in the monad) when passed an empty set.
758 minView :: Monad m => Set a -> m (a, Set a)
759 minView Tip = fail "Set.minView: empty set"
760 minView x = return (deleteFindMin x)
762 -- | /O(log n)/. Retrieves the maximal key of the set, and the set stripped from that element
763 -- @fail@s (in the monad) when passed an empty set.
764 maxView :: Monad m => Set a -> m (a, Set a)
765 maxView Tip = fail "Set.maxView: empty set"
766 maxView x = return (deleteFindMax x)
769 {--------------------------------------------------------------------
770 [balance x l r] balances two trees with value x.
771 The sizes of the trees should balance after decreasing the
772 size of one of them. (a rotation).
774 [delta] is the maximal relative difference between the sizes of
775 two trees, it corresponds with the [w] in Adams' paper,
776 or equivalently, [1/delta] corresponds with the $\alpha$
777 in Nievergelt's paper. Adams shows that [delta] should
778 be larger than 3.745 in order to garantee that the
779 rotations can always restore balance.
781 [ratio] is the ratio between an outer and inner sibling of the
782 heavier subtree in an unbalanced setting. It determines
783 whether a double or single rotation should be performed
784 to restore balance. It is correspondes with the inverse
785 of $\alpha$ in Adam's article.
788 - [delta] should be larger than 4.646 with a [ratio] of 2.
789 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
791 - A lower [delta] leads to a more 'perfectly' balanced tree.
792 - A higher [delta] performs less rebalancing.
794 - Balancing is automatic for random data and a balancing
795 scheme is only necessary to avoid pathological worst cases.
796 Almost any choice will do in practice
798 - Allthough it seems that a rather large [delta] may perform better
799 than smaller one, measurements have shown that the smallest [delta]
800 of 4 is actually the fastest on a wide range of operations. It
801 especially improves performance on worst-case scenarios like
802 a sequence of ordered insertions.
804 Note: in contrast to Adams' paper, we use a ratio of (at least) 2
805 to decide whether a single or double rotation is needed. Allthough
806 he actually proves that this ratio is needed to maintain the
807 invariants, his implementation uses a (invalid) ratio of 1.
808 He is aware of the problem though since he has put a comment in his
809 original source code that he doesn't care about generating a
810 slightly inbalanced tree since it doesn't seem to matter in practice.
811 However (since we use quickcheck :-) we will stick to strictly balanced
813 --------------------------------------------------------------------}
818 balance :: a -> Set a -> Set a -> Set a
820 | sizeL + sizeR <= 1 = Bin sizeX x l r
821 | sizeR >= delta*sizeL = rotateL x l r
822 | sizeL >= delta*sizeR = rotateR x l r
823 | otherwise = Bin sizeX x l r
827 sizeX = sizeL + sizeR + 1
830 rotateL x l r@(Bin _ _ ly ry)
831 | size ly < ratio*size ry = singleL x l r
832 | otherwise = doubleL x l r
834 rotateR x l@(Bin _ _ ly ry) r
835 | size ry < ratio*size ly = singleR x l r
836 | otherwise = doubleR x l r
839 singleL x1 t1 (Bin _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3
840 singleR x1 (Bin _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3)
842 doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
843 doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
846 {--------------------------------------------------------------------
847 The bin constructor maintains the size of the tree
848 --------------------------------------------------------------------}
849 bin :: a -> Set a -> Set a -> Set a
851 = Bin (size l + size r + 1) x l r
854 {--------------------------------------------------------------------
856 --------------------------------------------------------------------}
860 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
863 {--------------------------------------------------------------------
865 --------------------------------------------------------------------}
866 -- | /O(n)/. Show the tree that implements the set. The tree is shown
867 -- in a compressed, hanging format.
868 showTree :: Show a => Set a -> String
870 = showTreeWith True False s
873 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
874 the tree that implements the set. If @hang@ is
875 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
876 @wide@ is 'True', an extra wide version is shown.
878 > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
885 > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
896 > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
908 showTreeWith :: Show a => Bool -> Bool -> Set a -> String
909 showTreeWith hang wide t
910 | hang = (showsTreeHang wide [] t) ""
911 | otherwise = (showsTree wide [] [] t) ""
913 showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
914 showsTree wide lbars rbars t
916 Tip -> showsBars lbars . showString "|\n"
918 -> showsBars lbars . shows x . showString "\n"
920 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
921 showWide wide rbars .
922 showsBars lbars . shows x . showString "\n" .
923 showWide wide lbars .
924 showsTree wide (withEmpty lbars) (withBar lbars) l
926 showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
927 showsTreeHang wide bars t
929 Tip -> showsBars bars . showString "|\n"
931 -> showsBars bars . shows x . showString "\n"
933 -> showsBars bars . shows x . showString "\n" .
935 showsTreeHang wide (withBar bars) l .
937 showsTreeHang wide (withEmpty bars) r
941 | wide = showString (concat (reverse bars)) . showString "|\n"
944 showsBars :: [String] -> ShowS
948 _ -> showString (concat (reverse (tail bars))) . showString node
951 withBar bars = "| ":bars
952 withEmpty bars = " ":bars
954 {--------------------------------------------------------------------
956 --------------------------------------------------------------------}
957 -- | /O(n)/. Test if the internal set structure is valid.
958 valid :: Ord a => Set a -> Bool
960 = balanced t && ordered t && validsize t
963 = bounded (const True) (const True) t
968 Bin sz x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r
970 balanced :: Set a -> Bool
974 Bin sz x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
975 balanced l && balanced r
979 = (realsize t == Just (size t))
984 Bin sz x l r -> case (realsize l,realsize r) of
985 (Just n,Just m) | n+m+1 == sz -> Just sz
989 {--------------------------------------------------------------------
991 --------------------------------------------------------------------}
992 testTree :: [Int] -> Set Int
993 testTree xs = fromList xs
994 test1 = testTree [1..20]
995 test2 = testTree [30,29..10]
996 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
998 {--------------------------------------------------------------------
1000 --------------------------------------------------------------------}
1005 { configMaxTest = 500
1006 , configMaxFail = 5000
1007 , configSize = \n -> (div n 2 + 3)
1008 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
1012 {--------------------------------------------------------------------
1013 Arbitrary, reasonably balanced trees
1014 --------------------------------------------------------------------}
1015 instance (Enum a) => Arbitrary (Set a) where
1016 arbitrary = sized (arbtree 0 maxkey)
1017 where maxkey = 10000
1019 arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a)
1021 | n <= 0 = return Tip
1022 | lo >= hi = return Tip
1023 | otherwise = do{ i <- choose (lo,hi)
1024 ; m <- choose (1,30)
1025 ; let (ml,mr) | m==(1::Int)= (1,2)
1029 ; l <- arbtree lo (i-1) (n `div` ml)
1030 ; r <- arbtree (i+1) hi (n `div` mr)
1031 ; return (bin (toEnum i) l r)
1035 {--------------------------------------------------------------------
1037 --------------------------------------------------------------------}
1038 forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property
1040 = forAll arbitrary $ \t ->
1041 -- classify (balanced t) "balanced" $
1042 classify (size t == 0) "empty" $
1043 classify (size t > 0 && size t <= 10) "small" $
1044 classify (size t > 10 && size t <= 64) "medium" $
1045 classify (size t > 64) "large" $
1048 forValidIntTree :: Testable a => (Set Int -> a) -> Property
1052 forValidUnitTree :: Testable a => (Set Int -> a) -> Property
1058 = forValidUnitTree $ \t -> valid t
1060 {--------------------------------------------------------------------
1061 Single, Insert, Delete
1062 --------------------------------------------------------------------}
1063 prop_Single :: Int -> Bool
1065 = (insert x empty == singleton x)
1067 prop_InsertValid :: Int -> Property
1069 = forValidUnitTree $ \t -> valid (insert k t)
1071 prop_InsertDelete :: Int -> Set Int -> Property
1072 prop_InsertDelete k t
1073 = not (member k t) ==> delete k (insert k t) == t
1075 prop_DeleteValid :: Int -> Property
1077 = forValidUnitTree $ \t ->
1078 valid (delete k (insert k t))
1080 {--------------------------------------------------------------------
1082 --------------------------------------------------------------------}
1083 prop_Join :: Int -> Property
1085 = forValidUnitTree $ \t ->
1086 let (l,r) = split x t
1087 in valid (join x l r)
1089 prop_Merge :: Int -> Property
1091 = forValidUnitTree $ \t ->
1092 let (l,r) = split x t
1093 in valid (merge l r)
1096 {--------------------------------------------------------------------
1098 --------------------------------------------------------------------}
1099 prop_UnionValid :: Property
1101 = forValidUnitTree $ \t1 ->
1102 forValidUnitTree $ \t2 ->
1105 prop_UnionInsert :: Int -> Set Int -> Bool
1106 prop_UnionInsert x t
1107 = union t (singleton x) == insert x t
1109 prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool
1110 prop_UnionAssoc t1 t2 t3
1111 = union t1 (union t2 t3) == union (union t1 t2) t3
1113 prop_UnionComm :: Set Int -> Set Int -> Bool
1114 prop_UnionComm t1 t2
1115 = (union t1 t2 == union t2 t1)
1119 = forValidUnitTree $ \t1 ->
1120 forValidUnitTree $ \t2 ->
1121 valid (difference t1 t2)
1123 prop_Diff :: [Int] -> [Int] -> Bool
1125 = toAscList (difference (fromList xs) (fromList ys))
1126 == List.sort ((List.\\) (nub xs) (nub ys))
1129 = forValidUnitTree $ \t1 ->
1130 forValidUnitTree $ \t2 ->
1131 valid (intersection t1 t2)
1133 prop_Int :: [Int] -> [Int] -> Bool
1135 = toAscList (intersection (fromList xs) (fromList ys))
1136 == List.sort (nub ((List.intersect) (xs) (ys)))
1138 {--------------------------------------------------------------------
1140 --------------------------------------------------------------------}
1142 = forAll (choose (5,100)) $ \n ->
1143 let xs = [0..n::Int]
1144 in fromAscList xs == fromList xs
1146 prop_List :: [Int] -> Bool
1148 = (sort (nub xs) == toList (fromList xs))