[project @ 2006-01-02 19:38:01 by jpbernardy]
[packages/containers.git] / Data / Set.hs
1 -----------------------------------------------------------------------------
2 -- |
3 -- Module : Data.Set
4 -- Copyright : (c) Daan Leijen 2002
5 -- License : BSD-style
6 -- Maintainer : libraries@haskell.org
7 -- Stability : provisional
8 -- Portability : portable
9 --
10 -- An efficient implementation of sets.
11 --
12 -- This module is intended to be imported @qualified@, to avoid name
13 -- clashes with "Prelude" functions. eg.
14 --
15 -- > import Data.Set as Set
16 --
17 -- The implementation of 'Set' is based on /size balanced/ binary trees (or
18 -- trees of /bounded balance/) as described by:
19 --
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>.
23 --
24 -- * J. Nievergelt and E.M. Reingold,
25 -- \"/Binary search trees of bounded balance/\",
26 -- SIAM journal of computing 2(1), March 1973.
27 --
28 -- Note that the implementation is /left-biased/ -- the elements of a
29 -- first argument are always preferred 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
32 -- equality.
33 -----------------------------------------------------------------------------
34
35 module Data.Set (
36 -- * Set type
37 Set -- instance Eq,Ord,Show,Read,Data,Typeable
38
39 -- * Operators
40 , (\\)
41
42 -- * Query
43 , null
44 , size
45 , member
46 , isSubsetOf
47 , isProperSubsetOf
48
49 -- * Construction
50 , empty
51 , singleton
52 , insert
53 , delete
54
55 -- * Combine
56 , union, unions
57 , difference
58 , intersection
59
60 -- * Filter
61 , filter
62 , partition
63 , split
64 , splitMember
65
66 -- * Map
67 , map
68 , mapMonotonic
69
70 -- * Fold
71 , fold
72
73 -- * Min\/Max
74 , findMin
75 , findMax
76 , deleteMin
77 , deleteMax
78 , deleteFindMin
79 , deleteFindMax
80
81 -- * Conversion
82
83 -- ** List
84 , elems
85 , toList
86 , fromList
87
88 -- ** Ordered list
89 , toAscList
90 , fromAscList
91 , fromDistinctAscList
92
93 -- * Debugging
94 , showTree
95 , showTreeWith
96 , valid
97
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
112 ) where
113
114 import Prelude hiding (filter,foldr,null,map)
115 import qualified Data.List as List
116 import Data.Monoid (Monoid(..))
117 import Data.Typeable
118 import Data.Foldable (Foldable(foldMap))
119
120 {-
121 -- just for testing
122 import QuickCheck
123 import List (nub,sort)
124 import qualified List
125 -}
126
127 #if __GLASGOW_HASKELL__
128 import Text.Read
129 import Data.Generics.Basics
130 import Data.Generics.Instances
131 #endif
132
133 {--------------------------------------------------------------------
134 Operators
135 --------------------------------------------------------------------}
136 infixl 9 \\ --
137
138 -- | /O(n+m)/. See 'difference'.
139 (\\) :: Ord a => Set a -> Set a -> Set a
140 m1 \\ m2 = difference m1 m2
141
142 {--------------------------------------------------------------------
143 Sets are size balanced trees
144 --------------------------------------------------------------------}
145 -- | A set of values @a@.
146 data Set a = Tip
147 | Bin {-# UNPACK #-} !Size a !(Set a) !(Set a)
148
149 type Size = Int
150
151 instance Ord a => Monoid (Set a) where
152 mempty = empty
153 mappend = union
154 mconcat = unions
155
156 instance Foldable Set where
157 foldMap f Tip = mempty
158 foldMap f (Bin _s k l r) = foldMap f l `mappend` f k `mappend` foldMap f r
159
160 #if __GLASGOW_HASKELL__
161
162 {--------------------------------------------------------------------
163 A Data instance
164 --------------------------------------------------------------------}
165
166 -- This instance preserves data abstraction at the cost of inefficiency.
167 -- We omit reflection services for the sake of data abstraction.
168
169 instance (Data a, Ord a) => Data (Set a) where
170 gfoldl f z set = z fromList `f` (toList set)
171 toConstr _ = error "toConstr"
172 gunfold _ _ = error "gunfold"
173 dataTypeOf _ = mkNorepType "Data.Set.Set"
174 dataCast1 = gcast1
175
176 #endif
177
178 {--------------------------------------------------------------------
179 Query
180 --------------------------------------------------------------------}
181 -- | /O(1)/. Is this the empty set?
182 null :: Set a -> Bool
183 null t
184 = case t of
185 Tip -> True
186 Bin sz x l r -> False
187
188 -- | /O(1)/. The number of elements in the set.
189 size :: Set a -> Int
190 size t
191 = case t of
192 Tip -> 0
193 Bin sz x l r -> sz
194
195 -- | /O(log n)/. Is the element in the set?
196 member :: Ord a => a -> Set a -> Bool
197 member x t
198 = case t of
199 Tip -> False
200 Bin sz y l r
201 -> case compare x y of
202 LT -> member x l
203 GT -> member x r
204 EQ -> True
205
206 {--------------------------------------------------------------------
207 Construction
208 --------------------------------------------------------------------}
209 -- | /O(1)/. The empty set.
210 empty :: Set a
211 empty
212 = Tip
213
214 -- | /O(1)/. Create a singleton set.
215 singleton :: a -> Set a
216 singleton x
217 = Bin 1 x Tip Tip
218
219 {--------------------------------------------------------------------
220 Insertion, Deletion
221 --------------------------------------------------------------------}
222 -- | /O(log n)/. Insert an element in a set.
223 -- If the set already contains an element equal to the given value,
224 -- it is replaced with the new value.
225 insert :: Ord a => a -> Set a -> Set a
226 insert x t
227 = case t of
228 Tip -> singleton x
229 Bin sz y l r
230 -> case compare x y of
231 LT -> balance y (insert x l) r
232 GT -> balance y l (insert x r)
233 EQ -> Bin sz x l r
234
235
236 -- | /O(log n)/. Delete an element from a set.
237 delete :: Ord a => a -> Set a -> Set a
238 delete x t
239 = case t of
240 Tip -> Tip
241 Bin sz y l r
242 -> case compare x y of
243 LT -> balance y (delete x l) r
244 GT -> balance y l (delete x r)
245 EQ -> glue l r
246
247 {--------------------------------------------------------------------
248 Subset
249 --------------------------------------------------------------------}
250 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
251 isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
252 isProperSubsetOf s1 s2
253 = (size s1 < size s2) && (isSubsetOf s1 s2)
254
255
256 -- | /O(n+m)/. Is this a subset?
257 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
258 isSubsetOf :: Ord a => Set a -> Set a -> Bool
259 isSubsetOf t1 t2
260 = (size t1 <= size t2) && (isSubsetOfX t1 t2)
261
262 isSubsetOfX Tip t = True
263 isSubsetOfX t Tip = False
264 isSubsetOfX (Bin _ x l r) t
265 = found && isSubsetOfX l lt && isSubsetOfX r gt
266 where
267 (lt,found,gt) = splitMember x t
268
269
270 {--------------------------------------------------------------------
271 Minimal, Maximal
272 --------------------------------------------------------------------}
273 -- | /O(log n)/. The minimal element of a set.
274 findMin :: Set a -> a
275 findMin (Bin _ x Tip r) = x
276 findMin (Bin _ x l r) = findMin l
277 findMin Tip = error "Set.findMin: empty set has no minimal element"
278
279 -- | /O(log n)/. The maximal element of a set.
280 findMax :: Set a -> a
281 findMax (Bin _ x l Tip) = x
282 findMax (Bin _ x l r) = findMax r
283 findMax Tip = error "Set.findMax: empty set has no maximal element"
284
285 -- | /O(log n)/. Delete the minimal element.
286 deleteMin :: Set a -> Set a
287 deleteMin (Bin _ x Tip r) = r
288 deleteMin (Bin _ x l r) = balance x (deleteMin l) r
289 deleteMin Tip = Tip
290
291 -- | /O(log n)/. Delete the maximal element.
292 deleteMax :: Set a -> Set a
293 deleteMax (Bin _ x l Tip) = l
294 deleteMax (Bin _ x l r) = balance x l (deleteMax r)
295 deleteMax Tip = Tip
296
297
298 {--------------------------------------------------------------------
299 Union.
300 --------------------------------------------------------------------}
301 -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@).
302 unions :: Ord a => [Set a] -> Set a
303 unions ts
304 = foldlStrict union empty ts
305
306
307 -- | /O(n+m)/. The union of two sets, preferring the first set when
308 -- equal elements are encountered.
309 -- The implementation uses the efficient /hedge-union/ algorithm.
310 -- Hedge-union is more efficient on (bigset `union` smallset).
311 union :: Ord a => Set a -> Set a -> Set a
312 union Tip t2 = t2
313 union t1 Tip = t1
314 union t1 t2 = hedgeUnion (const LT) (const GT) t1 t2
315
316 hedgeUnion cmplo cmphi t1 Tip
317 = t1
318 hedgeUnion cmplo cmphi Tip (Bin _ x l r)
319 = join x (filterGt cmplo l) (filterLt cmphi r)
320 hedgeUnion cmplo cmphi (Bin _ x l r) t2
321 = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2))
322 (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2))
323 where
324 cmpx y = compare x y
325
326 {--------------------------------------------------------------------
327 Difference
328 --------------------------------------------------------------------}
329 -- | /O(n+m)/. Difference of two sets.
330 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
331 difference :: Ord a => Set a -> Set a -> Set a
332 difference Tip t2 = Tip
333 difference t1 Tip = t1
334 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
335
336 hedgeDiff cmplo cmphi Tip t
337 = Tip
338 hedgeDiff cmplo cmphi (Bin _ x l r) Tip
339 = join x (filterGt cmplo l) (filterLt cmphi r)
340 hedgeDiff cmplo cmphi t (Bin _ x l r)
341 = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l)
342 (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r)
343 where
344 cmpx y = compare x y
345
346 {--------------------------------------------------------------------
347 Intersection
348 --------------------------------------------------------------------}
349 -- | /O(n+m)/. The intersection of two sets.
350 -- Elements of the result come from the first set.
351 intersection :: Ord a => Set a -> Set a -> Set a
352 intersection Tip t = Tip
353 intersection t Tip = Tip
354 intersection t1@(Bin s1 x1 l1 r1) t2@(Bin s2 x2 l2 r2) =
355 if s1 >= s2 then
356 let (lt,found,gt) = splitLookup x2 t1
357 tl = intersection lt l2
358 tr = intersection gt r2
359 in case found of
360 Just x -> join x tl tr
361 Nothing -> merge tl tr
362 else let (lt,found,gt) = splitMember x1 t2
363 tl = intersection l1 lt
364 tr = intersection r1 gt
365 in if found then join x1 tl tr
366 else merge tl tr
367
368 {--------------------------------------------------------------------
369 Filter and partition
370 --------------------------------------------------------------------}
371 -- | /O(n)/. Filter all elements that satisfy the predicate.
372 filter :: Ord a => (a -> Bool) -> Set a -> Set a
373 filter p Tip = Tip
374 filter p (Bin _ x l r)
375 | p x = join x (filter p l) (filter p r)
376 | otherwise = merge (filter p l) (filter p r)
377
378 -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
379 -- the predicate and one with all elements that don't satisfy the predicate.
380 -- See also 'split'.
381 partition :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a)
382 partition p Tip = (Tip,Tip)
383 partition p (Bin _ x l r)
384 | p x = (join x l1 r1,merge l2 r2)
385 | otherwise = (merge l1 r1,join x l2 r2)
386 where
387 (l1,l2) = partition p l
388 (r1,r2) = partition p r
389
390 {----------------------------------------------------------------------
391 Map
392 ----------------------------------------------------------------------}
393
394 -- | /O(n*log n)/.
395 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
396 --
397 -- It's worth noting that the size of the result may be smaller if,
398 -- for some @(x,y)@, @x \/= y && f x == f y@
399
400 map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b
401 map f = fromList . List.map f . toList
402
403 -- | /O(n)/. The
404 --
405 -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.
406 -- /The precondition is not checked./
407 -- Semi-formally, we have:
408 --
409 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
410 -- > ==> mapMonotonic f s == map f s
411 -- > where ls = toList s
412
413 mapMonotonic :: (a->b) -> Set a -> Set b
414 mapMonotonic f Tip = Tip
415 mapMonotonic f (Bin sz x l r) =
416 Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)
417
418
419 {--------------------------------------------------------------------
420 Fold
421 --------------------------------------------------------------------}
422 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
423 fold :: (a -> b -> b) -> b -> Set a -> b
424 fold f z s
425 = foldr f z s
426
427 -- | /O(n)/. Post-order fold.
428 foldr :: (a -> b -> b) -> b -> Set a -> b
429 foldr f z Tip = z
430 foldr f z (Bin _ x l r) = foldr f (f x (foldr f z r)) l
431
432 {--------------------------------------------------------------------
433 List variations
434 --------------------------------------------------------------------}
435 -- | /O(n)/. The elements of a set.
436 elems :: Set a -> [a]
437 elems s
438 = toList s
439
440 {--------------------------------------------------------------------
441 Lists
442 --------------------------------------------------------------------}
443 -- | /O(n)/. Convert the set to a list of elements.
444 toList :: Set a -> [a]
445 toList s
446 = toAscList s
447
448 -- | /O(n)/. Convert the set to an ascending list of elements.
449 toAscList :: Set a -> [a]
450 toAscList t
451 = foldr (:) [] t
452
453
454 -- | /O(n*log n)/. Create a set from a list of elements.
455 fromList :: Ord a => [a] -> Set a
456 fromList xs
457 = foldlStrict ins empty xs
458 where
459 ins t x = insert x t
460
461 {--------------------------------------------------------------------
462 Building trees from ascending/descending lists can be done in linear time.
463
464 Note that if [xs] is ascending that:
465 fromAscList xs == fromList xs
466 --------------------------------------------------------------------}
467 -- | /O(n)/. Build a set from an ascending list in linear time.
468 -- /The precondition (input list is ascending) is not checked./
469 fromAscList :: Eq a => [a] -> Set a
470 fromAscList xs
471 = fromDistinctAscList (combineEq xs)
472 where
473 -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
474 combineEq xs
475 = case xs of
476 [] -> []
477 [x] -> [x]
478 (x:xx) -> combineEq' x xx
479
480 combineEq' z [] = [z]
481 combineEq' z (x:xs)
482 | z==x = combineEq' z xs
483 | otherwise = z:combineEq' x xs
484
485
486 -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
487 -- /The precondition (input list is strictly ascending) is not checked./
488 fromDistinctAscList :: [a] -> Set a
489 fromDistinctAscList xs
490 = build const (length xs) xs
491 where
492 -- 1) use continutations so that we use heap space instead of stack space.
493 -- 2) special case for n==5 to build bushier trees.
494 build c 0 xs = c Tip xs
495 build c 5 xs = case xs of
496 (x1:x2:x3:x4:x5:xx)
497 -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx
498 build c n xs = seq nr $ build (buildR nr c) nl xs
499 where
500 nl = n `div` 2
501 nr = n - nl - 1
502
503 buildR n c l (x:ys) = build (buildB l x c) n ys
504 buildB l x c r zs = c (bin x l r) zs
505
506 {--------------------------------------------------------------------
507 Eq converts the set to a list. In a lazy setting, this
508 actually seems one of the faster methods to compare two trees
509 and it is certainly the simplest :-)
510 --------------------------------------------------------------------}
511 instance Eq a => Eq (Set a) where
512 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
513
514 {--------------------------------------------------------------------
515 Ord
516 --------------------------------------------------------------------}
517
518 instance Ord a => Ord (Set a) where
519 compare s1 s2 = compare (toAscList s1) (toAscList s2)
520
521 {--------------------------------------------------------------------
522 Show
523 --------------------------------------------------------------------}
524 instance Show a => Show (Set a) where
525 showsPrec p xs = showParen (p > 10) $
526 showString "fromList " . shows (toList xs)
527
528 showSet :: (Show a) => [a] -> ShowS
529 showSet []
530 = showString "{}"
531 showSet (x:xs)
532 = showChar '{' . shows x . showTail xs
533 where
534 showTail [] = showChar '}'
535 showTail (x:xs) = showChar ',' . shows x . showTail xs
536
537 {--------------------------------------------------------------------
538 Read
539 --------------------------------------------------------------------}
540 instance (Read a, Ord a) => Read (Set a) where
541 #ifdef __GLASGOW_HASKELL__
542 readPrec = parens $ prec 10 $ do
543 Ident "fromList" <- lexP
544 xs <- readPrec
545 return (fromList xs)
546
547 readListPrec = readListPrecDefault
548 #else
549 readsPrec p = readParen (p > 10) $ \ r -> do
550 ("fromList",s) <- lex r
551 (xs,t) <- reads s
552 return (fromList xs,t)
553 #endif
554
555 {--------------------------------------------------------------------
556 Typeable/Data
557 --------------------------------------------------------------------}
558
559 #include "Typeable.h"
560 INSTANCE_TYPEABLE1(Set,setTc,"Set")
561
562 {--------------------------------------------------------------------
563 Utility functions that return sub-ranges of the original
564 tree. Some functions take a comparison function as argument to
565 allow comparisons against infinite values. A function [cmplo x]
566 should be read as [compare lo x].
567
568 [trim cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
569 and [cmphi x == GT] for the value [x] of the root.
570 [filterGt cmp t] A tree where for all values [k]. [cmp k == LT]
571 [filterLt cmp t] A tree where for all values [k]. [cmp k == GT]
572
573 [split k t] Returns two trees [l] and [r] where all values
574 in [l] are <[k] and all keys in [r] are >[k].
575 [splitMember k t] Just like [split] but also returns whether [k]
576 was found in the tree.
577 --------------------------------------------------------------------}
578
579 {--------------------------------------------------------------------
580 [trim lo hi t] trims away all subtrees that surely contain no
581 values between the range [lo] to [hi]. The returned tree is either
582 empty or the key of the root is between @lo@ and @hi@.
583 --------------------------------------------------------------------}
584 trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a
585 trim cmplo cmphi Tip = Tip
586 trim cmplo cmphi t@(Bin sx x l r)
587 = case cmplo x of
588 LT -> case cmphi x of
589 GT -> t
590 le -> trim cmplo cmphi l
591 ge -> trim cmplo cmphi r
592
593 trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a)
594 trimMemberLo lo cmphi Tip = (False,Tip)
595 trimMemberLo lo cmphi t@(Bin sx x l r)
596 = case compare lo x of
597 LT -> case cmphi x of
598 GT -> (member lo t, t)
599 le -> trimMemberLo lo cmphi l
600 GT -> trimMemberLo lo cmphi r
601 EQ -> (True,trim (compare lo) cmphi r)
602
603
604 {--------------------------------------------------------------------
605 [filterGt x t] filter all values >[x] from tree [t]
606 [filterLt x t] filter all values <[x] from tree [t]
607 --------------------------------------------------------------------}
608 filterGt :: (a -> Ordering) -> Set a -> Set a
609 filterGt cmp Tip = Tip
610 filterGt cmp (Bin sx x l r)
611 = case cmp x of
612 LT -> join x (filterGt cmp l) r
613 GT -> filterGt cmp r
614 EQ -> r
615
616 filterLt :: (a -> Ordering) -> Set a -> Set a
617 filterLt cmp Tip = Tip
618 filterLt cmp (Bin sx x l r)
619 = case cmp x of
620 LT -> filterLt cmp l
621 GT -> join x l (filterLt cmp r)
622 EQ -> l
623
624
625 {--------------------------------------------------------------------
626 Split
627 --------------------------------------------------------------------}
628 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
629 -- where all elements in @set1@ are lower than @x@ and all elements in
630 -- @set2@ larger than @x@. @x@ is not found in neither @set1@ nor @set2@.
631 split :: Ord a => a -> Set a -> (Set a,Set a)
632 split x Tip = (Tip,Tip)
633 split x (Bin sy y l r)
634 = case compare x y of
635 LT -> let (lt,gt) = split x l in (lt,join y gt r)
636 GT -> let (lt,gt) = split x r in (join y l lt,gt)
637 EQ -> (l,r)
638
639 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
640 -- element was found in the original set.
641 splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a)
642 splitMember x t = let (l,m,r) = splitLookup x t in
643 (l,maybe False (const True) m,r)
644
645 -- | /O(log n)/. Performs a 'split' but also returns the pivot
646 -- element that was found in the original set.
647 splitLookup :: Ord a => a -> Set a -> (Set a,Maybe a,Set a)
648 splitLookup x Tip = (Tip,Nothing,Tip)
649 splitLookup x (Bin sy y l r)
650 = case compare x y of
651 LT -> let (lt,found,gt) = splitLookup x l in (lt,found,join y gt r)
652 GT -> let (lt,found,gt) = splitLookup x r in (join y l lt,found,gt)
653 EQ -> (l,Just y,r)
654
655 {--------------------------------------------------------------------
656 Utility functions that maintain the balance properties of the tree.
657 All constructors assume that all values in [l] < [x] and all values
658 in [r] > [x], and that [l] and [r] are valid trees.
659
660 In order of sophistication:
661 [Bin sz x l r] The type constructor.
662 [bin x l r] Maintains the correct size, assumes that both [l]
663 and [r] are balanced with respect to each other.
664 [balance x l r] Restores the balance and size.
665 Assumes that the original tree was balanced and
666 that [l] or [r] has changed by at most one element.
667 [join x l r] Restores balance and size.
668
669 Furthermore, we can construct a new tree from two trees. Both operations
670 assume that all values in [l] < all values in [r] and that [l] and [r]
671 are valid:
672 [glue l r] Glues [l] and [r] together. Assumes that [l] and
673 [r] are already balanced with respect to each other.
674 [merge l r] Merges two trees and restores balance.
675
676 Note: in contrast to Adam's paper, we use (<=) comparisons instead
677 of (<) comparisons in [join], [merge] and [balance].
678 Quickcheck (on [difference]) showed that this was necessary in order
679 to maintain the invariants. It is quite unsatisfactory that I haven't
680 been able to find out why this is actually the case! Fortunately, it
681 doesn't hurt to be a bit more conservative.
682 --------------------------------------------------------------------}
683
684 {--------------------------------------------------------------------
685 Join
686 --------------------------------------------------------------------}
687 join :: a -> Set a -> Set a -> Set a
688 join x Tip r = insertMin x r
689 join x l Tip = insertMax x l
690 join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
691 | delta*sizeL <= sizeR = balance z (join x l lz) rz
692 | delta*sizeR <= sizeL = balance y ly (join x ry r)
693 | otherwise = bin x l r
694
695
696 -- insertMin and insertMax don't perform potentially expensive comparisons.
697 insertMax,insertMin :: a -> Set a -> Set a
698 insertMax x t
699 = case t of
700 Tip -> singleton x
701 Bin sz y l r
702 -> balance y l (insertMax x r)
703
704 insertMin x t
705 = case t of
706 Tip -> singleton x
707 Bin sz y l r
708 -> balance y (insertMin x l) r
709
710 {--------------------------------------------------------------------
711 [merge l r]: merges two trees.
712 --------------------------------------------------------------------}
713 merge :: Set a -> Set a -> Set a
714 merge Tip r = r
715 merge l Tip = l
716 merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
717 | delta*sizeL <= sizeR = balance y (merge l ly) ry
718 | delta*sizeR <= sizeL = balance x lx (merge rx r)
719 | otherwise = glue l r
720
721 {--------------------------------------------------------------------
722 [glue l r]: glues two trees together.
723 Assumes that [l] and [r] are already balanced with respect to each other.
724 --------------------------------------------------------------------}
725 glue :: Set a -> Set a -> Set a
726 glue Tip r = r
727 glue l Tip = l
728 glue l r
729 | size l > size r = let (m,l') = deleteFindMax l in balance m l' r
730 | otherwise = let (m,r') = deleteFindMin r in balance m l r'
731
732
733 -- | /O(log n)/. Delete and find the minimal element.
734 --
735 -- > deleteFindMin set = (findMin set, deleteMin set)
736
737 deleteFindMin :: Set a -> (a,Set a)
738 deleteFindMin t
739 = case t of
740 Bin _ x Tip r -> (x,r)
741 Bin _ x l r -> let (xm,l') = deleteFindMin l in (xm,balance x l' r)
742 Tip -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)
743
744 -- | /O(log n)/. Delete and find the maximal element.
745 --
746 -- > deleteFindMax set = (findMax set, deleteMax set)
747 deleteFindMax :: Set a -> (a,Set a)
748 deleteFindMax t
749 = case t of
750 Bin _ x l Tip -> (x,l)
751 Bin _ x l r -> let (xm,r') = deleteFindMax r in (xm,balance x l r')
752 Tip -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)
753
754
755 {--------------------------------------------------------------------
756 [balance x l r] balances two trees with value x.
757 The sizes of the trees should balance after decreasing the
758 size of one of them. (a rotation).
759
760 [delta] is the maximal relative difference between the sizes of
761 two trees, it corresponds with the [w] in Adams' paper,
762 or equivalently, [1/delta] corresponds with the $\alpha$
763 in Nievergelt's paper. Adams shows that [delta] should
764 be larger than 3.745 in order to garantee that the
765 rotations can always restore balance.
766
767 [ratio] is the ratio between an outer and inner sibling of the
768 heavier subtree in an unbalanced setting. It determines
769 whether a double or single rotation should be performed
770 to restore balance. It is correspondes with the inverse
771 of $\alpha$ in Adam's article.
772
773 Note that:
774 - [delta] should be larger than 4.646 with a [ratio] of 2.
775 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
776
777 - A lower [delta] leads to a more 'perfectly' balanced tree.
778 - A higher [delta] performs less rebalancing.
779
780 - Balancing is automatic for random data and a balancing
781 scheme is only necessary to avoid pathological worst cases.
782 Almost any choice will do in practice
783
784 - Allthough it seems that a rather large [delta] may perform better
785 than smaller one, measurements have shown that the smallest [delta]
786 of 4 is actually the fastest on a wide range of operations. It
787 especially improves performance on worst-case scenarios like
788 a sequence of ordered insertions.
789
790 Note: in contrast to Adams' paper, we use a ratio of (at least) 2
791 to decide whether a single or double rotation is needed. Allthough
792 he actually proves that this ratio is needed to maintain the
793 invariants, his implementation uses a (invalid) ratio of 1.
794 He is aware of the problem though since he has put a comment in his
795 original source code that he doesn't care about generating a
796 slightly inbalanced tree since it doesn't seem to matter in practice.
797 However (since we use quickcheck :-) we will stick to strictly balanced
798 trees.
799 --------------------------------------------------------------------}
800 delta,ratio :: Int
801 delta = 4
802 ratio = 2
803
804 balance :: a -> Set a -> Set a -> Set a
805 balance x l r
806 | sizeL + sizeR <= 1 = Bin sizeX x l r
807 | sizeR >= delta*sizeL = rotateL x l r
808 | sizeL >= delta*sizeR = rotateR x l r
809 | otherwise = Bin sizeX x l r
810 where
811 sizeL = size l
812 sizeR = size r
813 sizeX = sizeL + sizeR + 1
814
815 -- rotate
816 rotateL x l r@(Bin _ _ ly ry)
817 | size ly < ratio*size ry = singleL x l r
818 | otherwise = doubleL x l r
819
820 rotateR x l@(Bin _ _ ly ry) r
821 | size ry < ratio*size ly = singleR x l r
822 | otherwise = doubleR x l r
823
824 -- basic rotations
825 singleL x1 t1 (Bin _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3
826 singleR x1 (Bin _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3)
827
828 doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
829 doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
830
831
832 {--------------------------------------------------------------------
833 The bin constructor maintains the size of the tree
834 --------------------------------------------------------------------}
835 bin :: a -> Set a -> Set a -> Set a
836 bin x l r
837 = Bin (size l + size r + 1) x l r
838
839
840 {--------------------------------------------------------------------
841 Utilities
842 --------------------------------------------------------------------}
843 foldlStrict f z xs
844 = case xs of
845 [] -> z
846 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
847
848
849 {--------------------------------------------------------------------
850 Debugging
851 --------------------------------------------------------------------}
852 -- | /O(n)/. Show the tree that implements the set. The tree is shown
853 -- in a compressed, hanging format.
854 showTree :: Show a => Set a -> String
855 showTree s
856 = showTreeWith True False s
857
858
859 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
860 the tree that implements the set. If @hang@ is
861 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
862 @wide@ is 'True', an extra wide version is shown.
863
864 > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
865 > 4
866 > +--2
867 > | +--1
868 > | +--3
869 > +--5
870 >
871 > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
872 > 4
873 > |
874 > +--2
875 > | |
876 > | +--1
877 > | |
878 > | +--3
879 > |
880 > +--5
881 >
882 > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
883 > +--5
884 > |
885 > 4
886 > |
887 > | +--3
888 > | |
889 > +--2
890 > |
891 > +--1
892
893 -}
894 showTreeWith :: Show a => Bool -> Bool -> Set a -> String
895 showTreeWith hang wide t
896 | hang = (showsTreeHang wide [] t) ""
897 | otherwise = (showsTree wide [] [] t) ""
898
899 showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
900 showsTree wide lbars rbars t
901 = case t of
902 Tip -> showsBars lbars . showString "|\n"
903 Bin sz x Tip Tip
904 -> showsBars lbars . shows x . showString "\n"
905 Bin sz x l r
906 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
907 showWide wide rbars .
908 showsBars lbars . shows x . showString "\n" .
909 showWide wide lbars .
910 showsTree wide (withEmpty lbars) (withBar lbars) l
911
912 showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
913 showsTreeHang wide bars t
914 = case t of
915 Tip -> showsBars bars . showString "|\n"
916 Bin sz x Tip Tip
917 -> showsBars bars . shows x . showString "\n"
918 Bin sz x l r
919 -> showsBars bars . shows x . showString "\n" .
920 showWide wide bars .
921 showsTreeHang wide (withBar bars) l .
922 showWide wide bars .
923 showsTreeHang wide (withEmpty bars) r
924
925
926 showWide wide bars
927 | wide = showString (concat (reverse bars)) . showString "|\n"
928 | otherwise = id
929
930 showsBars :: [String] -> ShowS
931 showsBars bars
932 = case bars of
933 [] -> id
934 _ -> showString (concat (reverse (tail bars))) . showString node
935
936 node = "+--"
937 withBar bars = "| ":bars
938 withEmpty bars = " ":bars
939
940 {--------------------------------------------------------------------
941 Assertions
942 --------------------------------------------------------------------}
943 -- | /O(n)/. Test if the internal set structure is valid.
944 valid :: Ord a => Set a -> Bool
945 valid t
946 = balanced t && ordered t && validsize t
947
948 ordered t
949 = bounded (const True) (const True) t
950 where
951 bounded lo hi t
952 = case t of
953 Tip -> True
954 Bin sz x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r
955
956 balanced :: Set a -> Bool
957 balanced t
958 = case t of
959 Tip -> True
960 Bin sz x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
961 balanced l && balanced r
962
963
964 validsize t
965 = (realsize t == Just (size t))
966 where
967 realsize t
968 = case t of
969 Tip -> Just 0
970 Bin sz x l r -> case (realsize l,realsize r) of
971 (Just n,Just m) | n+m+1 == sz -> Just sz
972 other -> Nothing
973
974 {-
975 {--------------------------------------------------------------------
976 Testing
977 --------------------------------------------------------------------}
978 testTree :: [Int] -> Set Int
979 testTree xs = fromList xs
980 test1 = testTree [1..20]
981 test2 = testTree [30,29..10]
982 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
983
984 {--------------------------------------------------------------------
985 QuickCheck
986 --------------------------------------------------------------------}
987 qcheck prop
988 = check config prop
989 where
990 config = Config
991 { configMaxTest = 500
992 , configMaxFail = 5000
993 , configSize = \n -> (div n 2 + 3)
994 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
995 }
996
997
998 {--------------------------------------------------------------------
999 Arbitrary, reasonably balanced trees
1000 --------------------------------------------------------------------}
1001 instance (Enum a) => Arbitrary (Set a) where
1002 arbitrary = sized (arbtree 0 maxkey)
1003 where maxkey = 10000
1004
1005 arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a)
1006 arbtree lo hi n
1007 | n <= 0 = return Tip
1008 | lo >= hi = return Tip
1009 | otherwise = do{ i <- choose (lo,hi)
1010 ; m <- choose (1,30)
1011 ; let (ml,mr) | m==(1::Int)= (1,2)
1012 | m==2 = (2,1)
1013 | m==3 = (1,1)
1014 | otherwise = (2,2)
1015 ; l <- arbtree lo (i-1) (n `div` ml)
1016 ; r <- arbtree (i+1) hi (n `div` mr)
1017 ; return (bin (toEnum i) l r)
1018 }
1019
1020
1021 {--------------------------------------------------------------------
1022 Valid tree's
1023 --------------------------------------------------------------------}
1024 forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property
1025 forValid f
1026 = forAll arbitrary $ \t ->
1027 -- classify (balanced t) "balanced" $
1028 classify (size t == 0) "empty" $
1029 classify (size t > 0 && size t <= 10) "small" $
1030 classify (size t > 10 && size t <= 64) "medium" $
1031 classify (size t > 64) "large" $
1032 balanced t ==> f t
1033
1034 forValidIntTree :: Testable a => (Set Int -> a) -> Property
1035 forValidIntTree f
1036 = forValid f
1037
1038 forValidUnitTree :: Testable a => (Set Int -> a) -> Property
1039 forValidUnitTree f
1040 = forValid f
1041
1042
1043 prop_Valid
1044 = forValidUnitTree $ \t -> valid t
1045
1046 {--------------------------------------------------------------------
1047 Single, Insert, Delete
1048 --------------------------------------------------------------------}
1049 prop_Single :: Int -> Bool
1050 prop_Single x
1051 = (insert x empty == singleton x)
1052
1053 prop_InsertValid :: Int -> Property
1054 prop_InsertValid k
1055 = forValidUnitTree $ \t -> valid (insert k t)
1056
1057 prop_InsertDelete :: Int -> Set Int -> Property
1058 prop_InsertDelete k t
1059 = not (member k t) ==> delete k (insert k t) == t
1060
1061 prop_DeleteValid :: Int -> Property
1062 prop_DeleteValid k
1063 = forValidUnitTree $ \t ->
1064 valid (delete k (insert k t))
1065
1066 {--------------------------------------------------------------------
1067 Balance
1068 --------------------------------------------------------------------}
1069 prop_Join :: Int -> Property
1070 prop_Join x
1071 = forValidUnitTree $ \t ->
1072 let (l,r) = split x t
1073 in valid (join x l r)
1074
1075 prop_Merge :: Int -> Property
1076 prop_Merge x
1077 = forValidUnitTree $ \t ->
1078 let (l,r) = split x t
1079 in valid (merge l r)
1080
1081
1082 {--------------------------------------------------------------------
1083 Union
1084 --------------------------------------------------------------------}
1085 prop_UnionValid :: Property
1086 prop_UnionValid
1087 = forValidUnitTree $ \t1 ->
1088 forValidUnitTree $ \t2 ->
1089 valid (union t1 t2)
1090
1091 prop_UnionInsert :: Int -> Set Int -> Bool
1092 prop_UnionInsert x t
1093 = union t (singleton x) == insert x t
1094
1095 prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool
1096 prop_UnionAssoc t1 t2 t3
1097 = union t1 (union t2 t3) == union (union t1 t2) t3
1098
1099 prop_UnionComm :: Set Int -> Set Int -> Bool
1100 prop_UnionComm t1 t2
1101 = (union t1 t2 == union t2 t1)
1102
1103
1104 prop_DiffValid
1105 = forValidUnitTree $ \t1 ->
1106 forValidUnitTree $ \t2 ->
1107 valid (difference t1 t2)
1108
1109 prop_Diff :: [Int] -> [Int] -> Bool
1110 prop_Diff xs ys
1111 = toAscList (difference (fromList xs) (fromList ys))
1112 == List.sort ((List.\\) (nub xs) (nub ys))
1113
1114 prop_IntValid
1115 = forValidUnitTree $ \t1 ->
1116 forValidUnitTree $ \t2 ->
1117 valid (intersection t1 t2)
1118
1119 prop_Int :: [Int] -> [Int] -> Bool
1120 prop_Int xs ys
1121 = toAscList (intersection (fromList xs) (fromList ys))
1122 == List.sort (nub ((List.intersect) (xs) (ys)))
1123
1124 {--------------------------------------------------------------------
1125 Lists
1126 --------------------------------------------------------------------}
1127 prop_Ordered
1128 = forAll (choose (5,100)) $ \n ->
1129 let xs = [0..n::Int]
1130 in fromAscList xs == fromList xs
1131
1132 prop_List :: [Int] -> Bool
1133 prop_List xs
1134 = (sort (nub xs) == toList (fromList xs))
1135 -}
1136
1137 {--------------------------------------------------------------------
1138 Old Data.Set compatibility interface
1139 --------------------------------------------------------------------}
1140
1141 {-# DEPRECATED emptySet "Use empty instead" #-}
1142 -- | Obsolete equivalent of 'empty'.
1143 emptySet :: Set a
1144 emptySet = empty
1145
1146 {-# DEPRECATED mkSet "Use fromList instead" #-}
1147 -- | Obsolete equivalent of 'fromList'.
1148 mkSet :: Ord a => [a] -> Set a
1149 mkSet = fromList
1150
1151 {-# DEPRECATED setToList "Use elems instead." #-}
1152 -- | Obsolete equivalent of 'elems'.
1153 setToList :: Set a -> [a]
1154 setToList = elems
1155
1156 {-# DEPRECATED unitSet "Use singleton instead." #-}
1157 -- | Obsolete equivalent of 'singleton'.
1158 unitSet :: a -> Set a
1159 unitSet = singleton
1160
1161 {-# DEPRECATED elementOf "Use member instead." #-}
1162 -- | Obsolete equivalent of 'member'.
1163 elementOf :: Ord a => a -> Set a -> Bool
1164 elementOf = member
1165
1166 {-# DEPRECATED isEmptySet "Use null instead." #-}
1167 -- | Obsolete equivalent of 'null'.
1168 isEmptySet :: Set a -> Bool
1169 isEmptySet = null
1170
1171 {-# DEPRECATED cardinality "Use size instead." #-}
1172 -- | Obsolete equivalent of 'size'.
1173 cardinality :: Set a -> Int
1174 cardinality = size
1175
1176 {-# DEPRECATED unionManySets "Use unions instead." #-}
1177 -- | Obsolete equivalent of 'unions'.
1178 unionManySets :: Ord a => [Set a] -> Set a
1179 unionManySets = unions
1180
1181 {-# DEPRECATED minusSet "Use difference instead." #-}
1182 -- | Obsolete equivalent of 'difference'.
1183 minusSet :: Ord a => Set a -> Set a -> Set a
1184 minusSet = difference
1185
1186 {-# DEPRECATED mapSet "Use map instead." #-}
1187 -- | Obsolete equivalent of 'map'.
1188 mapSet :: (Ord a, Ord b) => (b -> a) -> Set b -> Set a
1189 mapSet = map
1190
1191 {-# DEPRECATED intersect "Use intersection instead." #-}
1192 -- | Obsolete equivalent of 'intersection'.
1193 intersect :: Ord a => Set a -> Set a -> Set a
1194 intersect = intersection
1195
1196 {-# DEPRECATED addToSet "Use 'flip insert' instead." #-}
1197 -- | Obsolete equivalent of @'flip' 'insert'@.
1198 addToSet :: Ord a => Set a -> a -> Set a
1199 addToSet = flip insert
1200
1201 {-# DEPRECATED delFromSet "Use `flip delete' instead." #-}
1202 -- | Obsolete equivalent of @'flip' 'delete'@.
1203 delFromSet :: Ord a => Set a -> a -> Set a
1204 delFromSet = flip delete