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 -- This module is intended to be imported @qualified@, to avoid name
13 -- clashes with "Prelude" functions. eg.
15 -- > import Data.Set as Set
17 -- The implementation of 'Set' is based on /size balanced/ binary trees (or
18 -- trees of /bounded balance/) as described by:
20 -- * Stephen Adams, \"/Efficient sets: a balancing act/\",
21 -- Journal of Functional Programming 3(4):553-562, October 1993,
22 -- <http://www.swiss.ai.mit.edu/~adams/BB>.
24 -- * J. Nievergelt and E.M. Reingold,
25 -- \"/Binary search trees of bounded balance/\",
26 -- SIAM journal of computing 2(1), March 1973.
28 -- Note that the implementation is /left-biased/ -- the elements of a
29 -- first argument are always perferred to the second, for example in
30 -- 'union' or 'insert'. Of course, left-biasing can only be observed
31 -- when equality is an equivalence relation instead of structural
33 -----------------------------------------------------------------------------
37 Set -- instance Eq,Show
98 -- * Old interface, DEPRECATED
99 ,emptySet, -- :: Set a
100 mkSet, -- :: Ord a => [a] -> Set a
101 setToList, -- :: Set a -> [a]
102 unitSet, -- :: a -> Set a
103 elementOf, -- :: Ord a => a -> Set a -> Bool
104 isEmptySet, -- :: Set a -> Bool
105 cardinality, -- :: Set a -> Int
106 unionManySets, -- :: Ord a => [Set a] -> Set a
107 minusSet, -- :: Ord a => Set a -> Set a -> Set a
108 mapSet, -- :: Ord a => (b -> a) -> Set b -> Set a
109 intersect, -- :: Ord a => Set a -> Set a -> Set a
110 addToSet, -- :: Ord a => Set a -> a -> Set a
111 delFromSet, -- :: Ord a => Set a -> a -> Set a
114 import Prelude hiding (filter,foldr,null,map)
115 import qualified Data.List as List
121 import List (nub,sort)
122 import qualified List
125 #if __GLASGOW_HASKELL__
126 import Data.Generics.Basics
127 import Data.Generics.Instances
130 {--------------------------------------------------------------------
132 --------------------------------------------------------------------}
135 -- | /O(n+m)/. See 'difference'.
136 (\\) :: Ord a => Set a -> Set a -> Set a
137 m1 \\ m2 = difference m1 m2
139 {--------------------------------------------------------------------
140 Sets are size balanced trees
141 --------------------------------------------------------------------}
142 -- | A set of values @a@.
144 | Bin {-# UNPACK #-} !Size a !(Set a) !(Set a)
148 #if __GLASGOW_HASKELL__
150 {--------------------------------------------------------------------
152 --------------------------------------------------------------------}
154 -- This instance preserves data abstraction at the cost of inefficiency.
155 -- We omit reflection services for the sake of data abstraction.
157 instance (Data a, Ord a) => Data (Set a) where
158 gfoldl f z set = z fromList `f` (toList set)
159 toConstr _ = error "toConstr"
160 gunfold _ _ = error "gunfold"
161 dataTypeOf _ = mkNorepType "Data.Set.Set"
165 {--------------------------------------------------------------------
167 --------------------------------------------------------------------}
168 -- | /O(1)/. Is this the empty set?
169 null :: Set a -> Bool
173 Bin sz x l r -> False
175 -- | /O(1)/. The number of elements in the set.
182 -- | /O(log n)/. Is the element in the set?
183 member :: Ord a => a -> Set a -> Bool
188 -> case compare x y of
193 {--------------------------------------------------------------------
195 --------------------------------------------------------------------}
196 -- | /O(1)/. The empty set.
201 -- | /O(1)/. Create a singleton set.
202 singleton :: a -> Set a
206 {--------------------------------------------------------------------
208 --------------------------------------------------------------------}
209 -- | /O(log n)/. Insert an element in a set.
210 insert :: Ord a => a -> Set a -> Set a
215 -> case compare x y of
216 LT -> balance y (insert x l) r
217 GT -> balance y l (insert x r)
221 -- | /O(log n)/. Delete an element from a set.
222 delete :: Ord a => a -> Set a -> Set a
227 -> case compare x y of
228 LT -> balance y (delete x l) r
229 GT -> balance y l (delete x r)
232 {--------------------------------------------------------------------
234 --------------------------------------------------------------------}
235 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
236 isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
237 isProperSubsetOf s1 s2
238 = (size s1 < size s2) && (isSubsetOf s1 s2)
241 -- | /O(n+m)/. Is this a subset?
242 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
243 isSubsetOf :: Ord a => Set a -> Set a -> Bool
245 = (size t1 <= size t2) && (isSubsetOfX t1 t2)
247 isSubsetOfX Tip t = True
248 isSubsetOfX t Tip = False
249 isSubsetOfX (Bin _ x l r) t
250 = found && isSubsetOfX l lt && isSubsetOfX r gt
252 (lt,found,gt) = splitMember x t
255 {--------------------------------------------------------------------
257 --------------------------------------------------------------------}
258 -- | /O(log n)/. The minimal element of a set.
259 findMin :: Set a -> a
260 findMin (Bin _ x Tip r) = x
261 findMin (Bin _ x l r) = findMin l
262 findMin Tip = error "Set.findMin: empty set has no minimal element"
264 -- | /O(log n)/. The maximal element of a set.
265 findMax :: Set a -> a
266 findMax (Bin _ x l Tip) = x
267 findMax (Bin _ x l r) = findMax r
268 findMax Tip = error "Set.findMax: empty set has no maximal element"
270 -- | /O(log n)/. Delete the minimal element.
271 deleteMin :: Set a -> Set a
272 deleteMin (Bin _ x Tip r) = r
273 deleteMin (Bin _ x l r) = balance x (deleteMin l) r
276 -- | /O(log n)/. Delete the maximal element.
277 deleteMax :: Set a -> Set a
278 deleteMax (Bin _ x l Tip) = l
279 deleteMax (Bin _ x l r) = balance x l (deleteMax r)
283 {--------------------------------------------------------------------
285 --------------------------------------------------------------------}
286 -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@).
287 unions :: Ord a => [Set a] -> Set a
289 = foldlStrict union empty ts
292 -- | /O(n+m)/. The union of two sets. Uses the efficient /hedge-union/ algorithm.
293 -- Hedge-union is more efficient on (bigset `union` smallset).
294 union :: Ord a => Set a -> Set a -> Set a
298 | size t1 >= size t2 = hedgeUnion (const LT) (const GT) t1 t2
299 | otherwise = hedgeUnion (const LT) (const GT) t2 t1
301 hedgeUnion cmplo cmphi t1 Tip
303 hedgeUnion cmplo cmphi Tip (Bin _ x l r)
304 = join x (filterGt cmplo l) (filterLt cmphi r)
305 hedgeUnion cmplo cmphi (Bin _ x l r) t2
306 = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2))
307 (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2))
311 {--------------------------------------------------------------------
313 --------------------------------------------------------------------}
314 -- | /O(n+m)/. Difference of two sets.
315 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
316 difference :: Ord a => Set a -> Set a -> Set a
317 difference Tip t2 = Tip
318 difference t1 Tip = t1
319 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
321 hedgeDiff cmplo cmphi Tip t
323 hedgeDiff cmplo cmphi (Bin _ x l r) Tip
324 = join x (filterGt cmplo l) (filterLt cmphi r)
325 hedgeDiff cmplo cmphi t (Bin _ x l r)
326 = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l)
327 (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r)
331 {--------------------------------------------------------------------
333 --------------------------------------------------------------------}
334 -- | /O(n+m)/. The intersection of two sets.
335 -- Intersection is more efficient on (bigset `intersection` smallset).
336 intersection :: Ord a => Set a -> Set a -> Set a
337 intersection Tip t = Tip
338 intersection t Tip = Tip
340 | size t1 >= size t2 = intersect' t1 t2
341 | otherwise = intersect' t2 t1
343 intersect' Tip t = Tip
344 intersect' t Tip = Tip
345 intersect' t (Bin _ x l r)
346 | found = join x tl tr
347 | otherwise = merge tl tr
349 (lt,found,gt) = splitMember x t
354 {--------------------------------------------------------------------
356 --------------------------------------------------------------------}
357 -- | /O(n)/. Filter all elements that satisfy the predicate.
358 filter :: Ord a => (a -> Bool) -> Set a -> Set a
360 filter p (Bin _ x l r)
361 | p x = join x (filter p l) (filter p r)
362 | otherwise = merge (filter p l) (filter p r)
364 -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
365 -- the predicate and one with all elements that don't satisfy the predicate.
367 partition :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a)
368 partition p Tip = (Tip,Tip)
369 partition p (Bin _ x l r)
370 | p x = (join x l1 r1,merge l2 r2)
371 | otherwise = (merge l1 r1,join x l2 r2)
373 (l1,l2) = partition p l
374 (r1,r2) = partition p r
376 {----------------------------------------------------------------------
378 ----------------------------------------------------------------------}
381 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
383 -- It's worth noting that the size of the result may be smaller if,
384 -- for some @(x,y)@, @x \/= y && f x == f y@
386 map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b
387 map f = fromList . List.map f . toList
391 -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.
392 -- /The precondition is not checked./
393 -- Semi-formally, we have:
395 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
396 -- > ==> mapMonotonic f s == map f s
397 -- > where ls = toList s
399 mapMonotonic :: (a->b) -> Set a -> Set b
400 mapMonotonic f Tip = Tip
401 mapMonotonic f (Bin sz x l r) =
402 Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)
405 {--------------------------------------------------------------------
407 --------------------------------------------------------------------}
408 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
409 fold :: (a -> b -> b) -> b -> Set a -> b
413 -- | /O(n)/. Post-order fold.
414 foldr :: (a -> b -> b) -> b -> Set a -> b
416 foldr f z (Bin _ x l r) = foldr f (f x (foldr f z r)) l
418 {--------------------------------------------------------------------
420 --------------------------------------------------------------------}
421 -- | /O(n)/. The elements of a set.
422 elems :: Set a -> [a]
426 {--------------------------------------------------------------------
428 --------------------------------------------------------------------}
429 -- | /O(n)/. Convert the set to a list of elements.
430 toList :: Set a -> [a]
434 -- | /O(n)/. Convert the set to an ascending list of elements.
435 toAscList :: Set a -> [a]
440 -- | /O(n*log n)/. Create a set from a list of elements.
441 fromList :: Ord a => [a] -> Set a
443 = foldlStrict ins empty xs
447 {--------------------------------------------------------------------
448 Building trees from ascending/descending lists can be done in linear time.
450 Note that if [xs] is ascending that:
451 fromAscList xs == fromList xs
452 --------------------------------------------------------------------}
453 -- | /O(n)/. Build a set from an ascending list in linear time.
454 -- /The precondition (input list is ascending) is not checked./
455 fromAscList :: Eq a => [a] -> Set a
457 = fromDistinctAscList (combineEq xs)
459 -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
464 (x:xx) -> combineEq' x xx
466 combineEq' z [] = [z]
468 | z==x = combineEq' z xs
469 | otherwise = z:combineEq' x xs
472 -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
473 -- /The precondition (input list is strictly ascending) is not checked./
474 fromDistinctAscList :: [a] -> Set a
475 fromDistinctAscList xs
476 = build const (length xs) xs
478 -- 1) use continutations so that we use heap space instead of stack space.
479 -- 2) special case for n==5 to build bushier trees.
480 build c 0 xs = c Tip xs
481 build c 5 xs = case xs of
483 -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx
484 build c n xs = seq nr $ build (buildR nr c) nl xs
489 buildR n c l (x:ys) = build (buildB l x c) n ys
490 buildB l x c r zs = c (bin x l r) zs
492 {--------------------------------------------------------------------
493 Eq converts the set to a list. In a lazy setting, this
494 actually seems one of the faster methods to compare two trees
495 and it is certainly the simplest :-)
496 --------------------------------------------------------------------}
497 instance Eq a => Eq (Set a) where
498 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
500 {--------------------------------------------------------------------
502 --------------------------------------------------------------------}
504 instance Ord a => Ord (Set a) where
505 compare s1 s2 = compare (toAscList s1) (toAscList s2)
507 {--------------------------------------------------------------------
509 --------------------------------------------------------------------}
510 instance Show a => Show (Set a) where
511 showsPrec d s = showSet (toAscList s)
513 showSet :: (Show a) => [a] -> ShowS
517 = showChar '{' . shows x . showTail xs
519 showTail [] = showChar '}'
520 showTail (x:xs) = showChar ',' . shows x . showTail xs
523 {--------------------------------------------------------------------
525 --------------------------------------------------------------------}
527 #include "Typeable.h"
528 INSTANCE_TYPEABLE1(Set,setTc,"Set")
530 {--------------------------------------------------------------------
531 Utility functions that return sub-ranges of the original
532 tree. Some functions take a comparison function as argument to
533 allow comparisons against infinite values. A function [cmplo x]
534 should be read as [compare lo x].
536 [trim cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
537 and [cmphi x == GT] for the value [x] of the root.
538 [filterGt cmp t] A tree where for all values [k]. [cmp k == LT]
539 [filterLt cmp t] A tree where for all values [k]. [cmp k == GT]
541 [split k t] Returns two trees [l] and [r] where all values
542 in [l] are <[k] and all keys in [r] are >[k].
543 [splitMember k t] Just like [split] but also returns whether [k]
544 was found in the tree.
545 --------------------------------------------------------------------}
547 {--------------------------------------------------------------------
548 [trim lo hi t] trims away all subtrees that surely contain no
549 values between the range [lo] to [hi]. The returned tree is either
550 empty or the key of the root is between @lo@ and @hi@.
551 --------------------------------------------------------------------}
552 trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a
553 trim cmplo cmphi Tip = Tip
554 trim cmplo cmphi t@(Bin sx x l r)
556 LT -> case cmphi x of
558 le -> trim cmplo cmphi l
559 ge -> trim cmplo cmphi r
561 trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a)
562 trimMemberLo lo cmphi Tip = (False,Tip)
563 trimMemberLo lo cmphi t@(Bin sx x l r)
564 = case compare lo x of
565 LT -> case cmphi x of
566 GT -> (member lo t, t)
567 le -> trimMemberLo lo cmphi l
568 GT -> trimMemberLo lo cmphi r
569 EQ -> (True,trim (compare lo) cmphi r)
572 {--------------------------------------------------------------------
573 [filterGt x t] filter all values >[x] from tree [t]
574 [filterLt x t] filter all values <[x] from tree [t]
575 --------------------------------------------------------------------}
576 filterGt :: (a -> Ordering) -> Set a -> Set a
577 filterGt cmp Tip = Tip
578 filterGt cmp (Bin sx x l r)
580 LT -> join x (filterGt cmp l) r
584 filterLt :: (a -> Ordering) -> Set a -> Set a
585 filterLt cmp Tip = Tip
586 filterLt cmp (Bin sx x l r)
589 GT -> join x l (filterLt cmp r)
593 {--------------------------------------------------------------------
595 --------------------------------------------------------------------}
596 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
597 -- where all elements in @set1@ are lower than @x@ and all elements in
598 -- @set2@ larger than @x@. @x@ is not found in neither @set1@ nor @set2@.
599 split :: Ord a => a -> Set a -> (Set a,Set a)
600 split x Tip = (Tip,Tip)
601 split x (Bin sy y l r)
602 = case compare x y of
603 LT -> let (lt,gt) = split x l in (lt,join y gt r)
604 GT -> let (lt,gt) = split x r in (join y l lt,gt)
607 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
608 -- element was found in the original set.
609 splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a)
610 splitMember x Tip = (Tip,False,Tip)
611 splitMember x (Bin sy y l r)
612 = case compare x y of
613 LT -> let (lt,found,gt) = splitMember x l in (lt,found,join y gt r)
614 GT -> let (lt,found,gt) = splitMember x r in (join y l lt,found,gt)
617 {--------------------------------------------------------------------
618 Utility functions that maintain the balance properties of the tree.
619 All constructors assume that all values in [l] < [x] and all values
620 in [r] > [x], and that [l] and [r] are valid trees.
622 In order of sophistication:
623 [Bin sz x l r] The type constructor.
624 [bin x l r] Maintains the correct size, assumes that both [l]
625 and [r] are balanced with respect to each other.
626 [balance x l r] Restores the balance and size.
627 Assumes that the original tree was balanced and
628 that [l] or [r] has changed by at most one element.
629 [join x l r] Restores balance and size.
631 Furthermore, we can construct a new tree from two trees. Both operations
632 assume that all values in [l] < all values in [r] and that [l] and [r]
634 [glue l r] Glues [l] and [r] together. Assumes that [l] and
635 [r] are already balanced with respect to each other.
636 [merge l r] Merges two trees and restores balance.
638 Note: in contrast to Adam's paper, we use (<=) comparisons instead
639 of (<) comparisons in [join], [merge] and [balance].
640 Quickcheck (on [difference]) showed that this was necessary in order
641 to maintain the invariants. It is quite unsatisfactory that I haven't
642 been able to find out why this is actually the case! Fortunately, it
643 doesn't hurt to be a bit more conservative.
644 --------------------------------------------------------------------}
646 {--------------------------------------------------------------------
648 --------------------------------------------------------------------}
649 join :: a -> Set a -> Set a -> Set a
650 join x Tip r = insertMin x r
651 join x l Tip = insertMax x l
652 join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
653 | delta*sizeL <= sizeR = balance z (join x l lz) rz
654 | delta*sizeR <= sizeL = balance y ly (join x ry r)
655 | otherwise = bin x l r
658 -- insertMin and insertMax don't perform potentially expensive comparisons.
659 insertMax,insertMin :: a -> Set a -> Set a
664 -> balance y l (insertMax x r)
670 -> balance y (insertMin x l) r
672 {--------------------------------------------------------------------
673 [merge l r]: merges two trees.
674 --------------------------------------------------------------------}
675 merge :: Set a -> Set a -> Set a
678 merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
679 | delta*sizeL <= sizeR = balance y (merge l ly) ry
680 | delta*sizeR <= sizeL = balance x lx (merge rx r)
681 | otherwise = glue l r
683 {--------------------------------------------------------------------
684 [glue l r]: glues two trees together.
685 Assumes that [l] and [r] are already balanced with respect to each other.
686 --------------------------------------------------------------------}
687 glue :: Set a -> Set a -> Set a
691 | size l > size r = let (m,l') = deleteFindMax l in balance m l' r
692 | otherwise = let (m,r') = deleteFindMin r in balance m l r'
695 -- | /O(log n)/. Delete and find the minimal element.
697 -- > deleteFindMin set = (findMin set, deleteMin set)
699 deleteFindMin :: Set a -> (a,Set a)
702 Bin _ x Tip r -> (x,r)
703 Bin _ x l r -> let (xm,l') = deleteFindMin l in (xm,balance x l' r)
704 Tip -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)
706 -- | /O(log n)/. Delete and find the maximal element.
708 -- > deleteFindMax set = (findMax set, deleteMax set)
709 deleteFindMax :: Set a -> (a,Set a)
712 Bin _ x l Tip -> (x,l)
713 Bin _ x l r -> let (xm,r') = deleteFindMax r in (xm,balance x l r')
714 Tip -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)
717 {--------------------------------------------------------------------
718 [balance x l r] balances two trees with value x.
719 The sizes of the trees should balance after decreasing the
720 size of one of them. (a rotation).
722 [delta] is the maximal relative difference between the sizes of
723 two trees, it corresponds with the [w] in Adams' paper,
724 or equivalently, [1/delta] corresponds with the $\alpha$
725 in Nievergelt's paper. Adams shows that [delta] should
726 be larger than 3.745 in order to garantee that the
727 rotations can always restore balance.
729 [ratio] is the ratio between an outer and inner sibling of the
730 heavier subtree in an unbalanced setting. It determines
731 whether a double or single rotation should be performed
732 to restore balance. It is correspondes with the inverse
733 of $\alpha$ in Adam's article.
736 - [delta] should be larger than 4.646 with a [ratio] of 2.
737 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
739 - A lower [delta] leads to a more 'perfectly' balanced tree.
740 - A higher [delta] performs less rebalancing.
742 - Balancing is automatic for random data and a balancing
743 scheme is only necessary to avoid pathological worst cases.
744 Almost any choice will do in practice
746 - Allthough it seems that a rather large [delta] may perform better
747 than smaller one, measurements have shown that the smallest [delta]
748 of 4 is actually the fastest on a wide range of operations. It
749 especially improves performance on worst-case scenarios like
750 a sequence of ordered insertions.
752 Note: in contrast to Adams' paper, we use a ratio of (at least) 2
753 to decide whether a single or double rotation is needed. Allthough
754 he actually proves that this ratio is needed to maintain the
755 invariants, his implementation uses a (invalid) ratio of 1.
756 He is aware of the problem though since he has put a comment in his
757 original source code that he doesn't care about generating a
758 slightly inbalanced tree since it doesn't seem to matter in practice.
759 However (since we use quickcheck :-) we will stick to strictly balanced
761 --------------------------------------------------------------------}
766 balance :: a -> Set a -> Set a -> Set a
768 | sizeL + sizeR <= 1 = Bin sizeX x l r
769 | sizeR >= delta*sizeL = rotateL x l r
770 | sizeL >= delta*sizeR = rotateR x l r
771 | otherwise = Bin sizeX x l r
775 sizeX = sizeL + sizeR + 1
778 rotateL x l r@(Bin _ _ ly ry)
779 | size ly < ratio*size ry = singleL x l r
780 | otherwise = doubleL x l r
782 rotateR x l@(Bin _ _ ly ry) r
783 | size ry < ratio*size ly = singleR x l r
784 | otherwise = doubleR x l r
787 singleL x1 t1 (Bin _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3
788 singleR x1 (Bin _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3)
790 doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
791 doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
794 {--------------------------------------------------------------------
795 The bin constructor maintains the size of the tree
796 --------------------------------------------------------------------}
797 bin :: a -> Set a -> Set a -> Set a
799 = Bin (size l + size r + 1) x l r
802 {--------------------------------------------------------------------
804 --------------------------------------------------------------------}
808 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
811 {--------------------------------------------------------------------
813 --------------------------------------------------------------------}
814 -- | /O(n)/. Show the tree that implements the set. The tree is shown
815 -- in a compressed, hanging format.
816 showTree :: Show a => Set a -> String
818 = showTreeWith True False s
821 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
822 the tree that implements the set. If @hang@ is
823 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
824 @wide@ is 'True', an extra wide version is shown.
826 > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
833 > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
844 > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
856 showTreeWith :: Show a => Bool -> Bool -> Set a -> String
857 showTreeWith hang wide t
858 | hang = (showsTreeHang wide [] t) ""
859 | otherwise = (showsTree wide [] [] t) ""
861 showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
862 showsTree wide lbars rbars t
864 Tip -> showsBars lbars . showString "|\n"
866 -> showsBars lbars . shows x . showString "\n"
868 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
869 showWide wide rbars .
870 showsBars lbars . shows x . showString "\n" .
871 showWide wide lbars .
872 showsTree wide (withEmpty lbars) (withBar lbars) l
874 showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
875 showsTreeHang wide bars t
877 Tip -> showsBars bars . showString "|\n"
879 -> showsBars bars . shows x . showString "\n"
881 -> showsBars bars . shows x . showString "\n" .
883 showsTreeHang wide (withBar bars) l .
885 showsTreeHang wide (withEmpty bars) r
889 | wide = showString (concat (reverse bars)) . showString "|\n"
892 showsBars :: [String] -> ShowS
896 _ -> showString (concat (reverse (tail bars))) . showString node
899 withBar bars = "| ":bars
900 withEmpty bars = " ":bars
902 {--------------------------------------------------------------------
904 --------------------------------------------------------------------}
905 -- | /O(n)/. Test if the internal set structure is valid.
906 valid :: Ord a => Set a -> Bool
908 = balanced t && ordered t && validsize t
911 = bounded (const True) (const True) t
916 Bin sz x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r
918 balanced :: Set a -> Bool
922 Bin sz x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
923 balanced l && balanced r
927 = (realsize t == Just (size t))
932 Bin sz x l r -> case (realsize l,realsize r) of
933 (Just n,Just m) | n+m+1 == sz -> Just sz
937 {--------------------------------------------------------------------
939 --------------------------------------------------------------------}
940 testTree :: [Int] -> Set Int
941 testTree xs = fromList xs
942 test1 = testTree [1..20]
943 test2 = testTree [30,29..10]
944 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
946 {--------------------------------------------------------------------
948 --------------------------------------------------------------------}
953 { configMaxTest = 500
954 , configMaxFail = 5000
955 , configSize = \n -> (div n 2 + 3)
956 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
960 {--------------------------------------------------------------------
961 Arbitrary, reasonably balanced trees
962 --------------------------------------------------------------------}
963 instance (Enum a) => Arbitrary (Set a) where
964 arbitrary = sized (arbtree 0 maxkey)
967 arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a)
969 | n <= 0 = return Tip
970 | lo >= hi = return Tip
971 | otherwise = do{ i <- choose (lo,hi)
973 ; let (ml,mr) | m==(1::Int)= (1,2)
977 ; l <- arbtree lo (i-1) (n `div` ml)
978 ; r <- arbtree (i+1) hi (n `div` mr)
979 ; return (bin (toEnum i) l r)
983 {--------------------------------------------------------------------
985 --------------------------------------------------------------------}
986 forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property
988 = forAll arbitrary $ \t ->
989 -- classify (balanced t) "balanced" $
990 classify (size t == 0) "empty" $
991 classify (size t > 0 && size t <= 10) "small" $
992 classify (size t > 10 && size t <= 64) "medium" $
993 classify (size t > 64) "large" $
996 forValidIntTree :: Testable a => (Set Int -> a) -> Property
1000 forValidUnitTree :: Testable a => (Set Int -> a) -> Property
1006 = forValidUnitTree $ \t -> valid t
1008 {--------------------------------------------------------------------
1009 Single, Insert, Delete
1010 --------------------------------------------------------------------}
1011 prop_Single :: Int -> Bool
1013 = (insert x empty == singleton x)
1015 prop_InsertValid :: Int -> Property
1017 = forValidUnitTree $ \t -> valid (insert k t)
1019 prop_InsertDelete :: Int -> Set Int -> Property
1020 prop_InsertDelete k t
1021 = not (member k t) ==> delete k (insert k t) == t
1023 prop_DeleteValid :: Int -> Property
1025 = forValidUnitTree $ \t ->
1026 valid (delete k (insert k t))
1028 {--------------------------------------------------------------------
1030 --------------------------------------------------------------------}
1031 prop_Join :: Int -> Property
1033 = forValidUnitTree $ \t ->
1034 let (l,r) = split x t
1035 in valid (join x l r)
1037 prop_Merge :: Int -> Property
1039 = forValidUnitTree $ \t ->
1040 let (l,r) = split x t
1041 in valid (merge l r)
1044 {--------------------------------------------------------------------
1046 --------------------------------------------------------------------}
1047 prop_UnionValid :: Property
1049 = forValidUnitTree $ \t1 ->
1050 forValidUnitTree $ \t2 ->
1053 prop_UnionInsert :: Int -> Set Int -> Bool
1054 prop_UnionInsert x t
1055 = union t (singleton x) == insert x t
1057 prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool
1058 prop_UnionAssoc t1 t2 t3
1059 = union t1 (union t2 t3) == union (union t1 t2) t3
1061 prop_UnionComm :: Set Int -> Set Int -> Bool
1062 prop_UnionComm t1 t2
1063 = (union t1 t2 == union t2 t1)
1067 = forValidUnitTree $ \t1 ->
1068 forValidUnitTree $ \t2 ->
1069 valid (difference t1 t2)
1071 prop_Diff :: [Int] -> [Int] -> Bool
1073 = toAscList (difference (fromList xs) (fromList ys))
1074 == List.sort ((List.\\) (nub xs) (nub ys))
1077 = forValidUnitTree $ \t1 ->
1078 forValidUnitTree $ \t2 ->
1079 valid (intersection t1 t2)
1081 prop_Int :: [Int] -> [Int] -> Bool
1083 = toAscList (intersection (fromList xs) (fromList ys))
1084 == List.sort (nub ((List.intersect) (xs) (ys)))
1086 {--------------------------------------------------------------------
1088 --------------------------------------------------------------------}
1090 = forAll (choose (5,100)) $ \n ->
1091 let xs = [0..n::Int]
1092 in fromAscList xs == fromList xs
1094 prop_List :: [Int] -> Bool
1096 = (sort (nub xs) == toList (fromList xs))
1099 {--------------------------------------------------------------------
1100 Old Data.Set compatibility interface
1101 --------------------------------------------------------------------}
1103 {-# DEPRECATED emptySet "Use empty instead" #-}
1104 -- | Obsolete equivalent of 'empty'.
1108 {-# DEPRECATED mkSet "Use fromList instead" #-}
1109 -- | Obsolete equivalent of 'fromList'.
1110 mkSet :: Ord a => [a] -> Set a
1113 {-# DEPRECATED setToList "Use elems instead." #-}
1114 -- | Obsolete equivalent of 'elems'.
1115 setToList :: Set a -> [a]
1118 {-# DEPRECATED unitSet "Use singleton instead." #-}
1119 -- | Obsolete equivalent of 'singleton'.
1120 unitSet :: a -> Set a
1123 {-# DEPRECATED elementOf "Use member instead." #-}
1124 -- | Obsolete equivalent of 'member'.
1125 elementOf :: Ord a => a -> Set a -> Bool
1128 {-# DEPRECATED isEmptySet "Use null instead." #-}
1129 -- | Obsolete equivalent of 'null'.
1130 isEmptySet :: Set a -> Bool
1133 {-# DEPRECATED cardinality "Use size instead." #-}
1134 -- | Obsolete equivalent of 'size'.
1135 cardinality :: Set a -> Int
1138 {-# DEPRECATED unionManySets "Use unions instead." #-}
1139 -- | Obsolete equivalent of 'unions'.
1140 unionManySets :: Ord a => [Set a] -> Set a
1141 unionManySets = unions
1143 {-# DEPRECATED minusSet "Use difference instead." #-}
1144 -- | Obsolete equivalent of 'difference'.
1145 minusSet :: Ord a => Set a -> Set a -> Set a
1146 minusSet = difference
1148 {-# DEPRECATED mapSet "Use map instead." #-}
1149 -- | Obsolete equivalent of 'map'.
1150 mapSet :: (Ord a, Ord b) => (b -> a) -> Set b -> Set a
1153 {-# DEPRECATED intersect "Use intersection instead." #-}
1154 -- | Obsolete equivalent of 'intersection'.
1155 intersect :: Ord a => Set a -> Set a -> Set a
1156 intersect = intersection
1158 {-# DEPRECATED addToSet "Use 'flip insert' instead." #-}
1159 -- | Obsolete equivalent of @'flip' 'insert'@.
1160 addToSet :: Ord a => Set a -> a -> Set a
1161 addToSet = flip insert
1163 {-# DEPRECATED delFromSet "Use `flip delete' instead." #-}
1164 -- | Obsolete equivalent of @'flip' 'delete'@.
1165 delFromSet :: Ord a => Set a -> a -> Set a
1166 delFromSet = flip delete