[project @ 2005-01-20 19:00:26 by ross]
[packages/old-time.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 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
32 -- equality.
33 -----------------------------------------------------------------------------
34
35 module Data.Set (
36 -- * Set type
37 Set -- instance Eq,Show
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 Data.Monoid
116 import qualified Data.List as List
117 import Data.Typeable
118
119 {-
120 -- just for testing
121 import QuickCheck
122 import List (nub,sort)
123 import qualified List
124 -}
125
126 #if __GLASGOW_HASKELL__
127 import Data.Generics.Basics
128 import Data.Generics.Instances
129 #endif
130
131 {--------------------------------------------------------------------
132 Operators
133 --------------------------------------------------------------------}
134 infixl 9 \\ --
135
136 -- | /O(n+m)/. See 'difference'.
137 (\\) :: Ord a => Set a -> Set a -> Set a
138 m1 \\ m2 = difference m1 m2
139
140 {--------------------------------------------------------------------
141 Sets are size balanced trees
142 --------------------------------------------------------------------}
143 -- | A set of values @a@.
144 data Set a = Tip
145 | Bin {-# UNPACK #-} !Size a !(Set a) !(Set a)
146
147 type Size = Int
148
149 #if __GLASGOW_HASKELL__
150
151 {--------------------------------------------------------------------
152 A Data instance
153 --------------------------------------------------------------------}
154
155 -- This instance preserves data abstraction at the cost of inefficiency.
156 -- We omit reflection services for the sake of data abstraction.
157
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
164 #endif
165
166 {--------------------------------------------------------------------
167 Query
168 --------------------------------------------------------------------}
169 -- | /O(1)/. Is this the empty set?
170 null :: Set a -> Bool
171 null t
172 = case t of
173 Tip -> True
174 Bin sz x l r -> False
175
176 -- | /O(1)/. The number of elements in the set.
177 size :: Set a -> Int
178 size t
179 = case t of
180 Tip -> 0
181 Bin sz x l r -> sz
182
183 -- | /O(log n)/. Is the element in the set?
184 member :: Ord a => a -> Set a -> Bool
185 member x t
186 = case t of
187 Tip -> False
188 Bin sz y l r
189 -> case compare x y of
190 LT -> member x l
191 GT -> member x r
192 EQ -> True
193
194 {--------------------------------------------------------------------
195 Construction
196 --------------------------------------------------------------------}
197 -- | /O(1)/. The empty set.
198 empty :: Set a
199 empty
200 = Tip
201
202 -- | /O(1)/. Create a singleton set.
203 singleton :: a -> Set a
204 singleton x
205 = Bin 1 x Tip Tip
206
207 {--------------------------------------------------------------------
208 Insertion, Deletion
209 --------------------------------------------------------------------}
210 -- | /O(log n)/. Insert an element in a set.
211 insert :: Ord a => a -> Set a -> Set a
212 insert x t
213 = case t of
214 Tip -> singleton x
215 Bin sz y l r
216 -> case compare x y of
217 LT -> balance y (insert x l) r
218 GT -> balance y l (insert x r)
219 EQ -> Bin sz x l r
220
221
222 -- | /O(log n)/. Delete an element from a set.
223 delete :: Ord a => a -> Set a -> Set a
224 delete x t
225 = case t of
226 Tip -> Tip
227 Bin sz y l r
228 -> case compare x y of
229 LT -> balance y (delete x l) r
230 GT -> balance y l (delete x r)
231 EQ -> glue l r
232
233 {--------------------------------------------------------------------
234 Subset
235 --------------------------------------------------------------------}
236 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
237 isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
238 isProperSubsetOf s1 s2
239 = (size s1 < size s2) && (isSubsetOf s1 s2)
240
241
242 -- | /O(n+m)/. Is this a subset?
243 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
244 isSubsetOf :: Ord a => Set a -> Set a -> Bool
245 isSubsetOf t1 t2
246 = (size t1 <= size t2) && (isSubsetOfX t1 t2)
247
248 isSubsetOfX Tip t = True
249 isSubsetOfX t Tip = False
250 isSubsetOfX (Bin _ x l r) t
251 = found && isSubsetOfX l lt && isSubsetOfX r gt
252 where
253 (found,lt,gt) = splitMember x t
254
255
256 {--------------------------------------------------------------------
257 Minimal, Maximal
258 --------------------------------------------------------------------}
259 -- | /O(log n)/. The minimal element of a set.
260 findMin :: Set a -> a
261 findMin (Bin _ x Tip r) = x
262 findMin (Bin _ x l r) = findMin l
263 findMin Tip = error "Set.findMin: empty set has no minimal element"
264
265 -- | /O(log n)/. The maximal element of a set.
266 findMax :: Set a -> a
267 findMax (Bin _ x l Tip) = x
268 findMax (Bin _ x l r) = findMax r
269 findMax Tip = error "Set.findMax: empty set has no maximal element"
270
271 -- | /O(log n)/. Delete the minimal element.
272 deleteMin :: Set a -> Set a
273 deleteMin (Bin _ x Tip r) = r
274 deleteMin (Bin _ x l r) = balance x (deleteMin l) r
275 deleteMin Tip = Tip
276
277 -- | /O(log n)/. Delete the maximal element.
278 deleteMax :: Set a -> Set a
279 deleteMax (Bin _ x l Tip) = l
280 deleteMax (Bin _ x l r) = balance x l (deleteMax r)
281 deleteMax Tip = Tip
282
283
284 {--------------------------------------------------------------------
285 Union.
286 --------------------------------------------------------------------}
287 -- | The union of a list of sets: (@'unions' == 'foldl' 'union' 'empty'@).
288 unions :: Ord a => [Set a] -> Set a
289 unions ts
290 = foldlStrict union empty ts
291
292
293 -- | /O(n+m)/. The union of two sets. Uses the efficient /hedge-union/ algorithm.
294 -- Hedge-union is more efficient on (bigset `union` smallset).
295 union :: Ord a => Set a -> Set a -> Set a
296 union Tip t2 = t2
297 union t1 Tip = t1
298 union t1 t2
299 | size t1 >= size t2 = hedgeUnion (const LT) (const GT) t1 t2
300 | otherwise = hedgeUnion (const LT) (const GT) t2 t1
301
302 hedgeUnion cmplo cmphi t1 Tip
303 = t1
304 hedgeUnion cmplo cmphi Tip (Bin _ x l r)
305 = join x (filterGt cmplo l) (filterLt cmphi r)
306 hedgeUnion cmplo cmphi (Bin _ x l r) t2
307 = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2))
308 (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2))
309 where
310 cmpx y = compare x y
311
312 {--------------------------------------------------------------------
313 Difference
314 --------------------------------------------------------------------}
315 -- | /O(n+m)/. Difference of two sets.
316 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
317 difference :: Ord a => Set a -> Set a -> Set a
318 difference Tip t2 = Tip
319 difference t1 Tip = t1
320 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
321
322 hedgeDiff cmplo cmphi Tip t
323 = Tip
324 hedgeDiff cmplo cmphi (Bin _ x l r) Tip
325 = join x (filterGt cmplo l) (filterLt cmphi r)
326 hedgeDiff cmplo cmphi t (Bin _ x l r)
327 = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l)
328 (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r)
329 where
330 cmpx y = compare x y
331
332 {--------------------------------------------------------------------
333 Intersection
334 --------------------------------------------------------------------}
335 -- | /O(n+m)/. The intersection of two sets.
336 -- Intersection is more efficient on (bigset `intersection` smallset).
337 intersection :: Ord a => Set a -> Set a -> Set a
338 intersection Tip t = Tip
339 intersection t Tip = Tip
340 intersection t1 t2
341 | size t1 >= size t2 = intersect' t1 t2
342 | otherwise = intersect' t2 t1
343
344 intersect' Tip t = Tip
345 intersect' t Tip = Tip
346 intersect' t (Bin _ x l r)
347 | found = join x tl tr
348 | otherwise = merge tl tr
349 where
350 (found,lt,gt) = splitMember x t
351 tl = intersect' lt l
352 tr = intersect' gt r
353
354
355 {--------------------------------------------------------------------
356 Filter and partition
357 --------------------------------------------------------------------}
358 -- | /O(n)/. Filter all elements that satisfy the predicate.
359 filter :: Ord a => (a -> Bool) -> Set a -> Set a
360 filter p Tip = Tip
361 filter p (Bin _ x l r)
362 | p x = join x (filter p l) (filter p r)
363 | otherwise = merge (filter p l) (filter p r)
364
365 -- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
366 -- the predicate and one with all elements that don't satisfy the predicate.
367 -- See also 'split'.
368 partition :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a)
369 partition p Tip = (Tip,Tip)
370 partition p (Bin _ x l r)
371 | p x = (join x l1 r1,merge l2 r2)
372 | otherwise = (merge l1 r1,join x l2 r2)
373 where
374 (l1,l2) = partition p l
375 (r1,r2) = partition p r
376
377 {----------------------------------------------------------------------
378 Map
379 ----------------------------------------------------------------------}
380
381 -- | /O(n*log n)/.
382 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
383 --
384 -- It's worth noting that the size of the result may be smaller if,
385 -- for some @(x,y)@, @x \/= y && f x == f y@
386
387 map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b
388 map f = fromList . List.map f . toList
389
390 -- | /O(n)/. The
391 --
392 -- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.
393 -- /The precondition is not checked./
394 -- Semi-formally, we have:
395 --
396 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
397 -- > ==> mapMonotonic f s == map f s
398 -- > where ls = toList s
399
400 mapMonotonic :: (a->b) -> Set a -> Set b
401 mapMonotonic f Tip = Tip
402 mapMonotonic f (Bin sz x l r) =
403 Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)
404
405
406 {--------------------------------------------------------------------
407 Fold
408 --------------------------------------------------------------------}
409 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
410 fold :: (a -> b -> b) -> b -> Set a -> b
411 fold f z s
412 = foldr f z s
413
414 -- | /O(n)/. Post-order fold.
415 foldr :: (a -> b -> b) -> b -> Set a -> b
416 foldr f z Tip = z
417 foldr f z (Bin _ x l r) = foldr f (f x (foldr f z r)) l
418
419 {--------------------------------------------------------------------
420 List variations
421 --------------------------------------------------------------------}
422 -- | /O(n)/. The elements of a set.
423 elems :: Set a -> [a]
424 elems s
425 = toList s
426
427 {--------------------------------------------------------------------
428 Lists
429 --------------------------------------------------------------------}
430 -- | /O(n)/. Convert the set to an ascending list of elements.
431 toList :: Set a -> [a]
432 toList s
433 = toAscList s
434
435 -- | /O(n)/. Convert the set to an ascending list of elements.
436 toAscList :: Set a -> [a]
437 toAscList t
438 = foldr (:) [] t
439
440
441 -- | /O(n*log n)/. Create a set from a list of elements.
442 fromList :: Ord a => [a] -> Set a
443 fromList xs
444 = foldlStrict ins empty xs
445 where
446 ins t x = insert x t
447
448 {--------------------------------------------------------------------
449 Building trees from ascending/descending lists can be done in linear time.
450
451 Note that if [xs] is ascending that:
452 fromAscList xs == fromList xs
453 --------------------------------------------------------------------}
454 -- | /O(n)/. Build a set from an ascending list in linear time.
455 -- /The precondition (input list is ascending) is not checked./
456 fromAscList :: Eq a => [a] -> Set a
457 fromAscList xs
458 = fromDistinctAscList (combineEq xs)
459 where
460 -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
461 combineEq xs
462 = case xs of
463 [] -> []
464 [x] -> [x]
465 (x:xx) -> combineEq' x xx
466
467 combineEq' z [] = [z]
468 combineEq' z (x:xs)
469 | z==x = combineEq' z xs
470 | otherwise = z:combineEq' x xs
471
472
473 -- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
474 -- /The precondition (input list is strictly ascending) is not checked./
475 fromDistinctAscList :: [a] -> Set a
476 fromDistinctAscList xs
477 = build const (length xs) xs
478 where
479 -- 1) use continutations so that we use heap space instead of stack space.
480 -- 2) special case for n==5 to build bushier trees.
481 build c 0 xs = c Tip xs
482 build c 5 xs = case xs of
483 (x1:x2:x3:x4:x5:xx)
484 -> c (bin x4 (bin x2 (singleton x1) (singleton x3)) (singleton x5)) xx
485 build c n xs = seq nr $ build (buildR nr c) nl xs
486 where
487 nl = n `div` 2
488 nr = n - nl - 1
489
490 buildR n c l (x:ys) = build (buildB l x c) n ys
491 buildB l x c r zs = c (bin x l r) zs
492
493 {--------------------------------------------------------------------
494 Eq converts the set to a list. In a lazy setting, this
495 actually seems one of the faster methods to compare two trees
496 and it is certainly the simplest :-)
497 --------------------------------------------------------------------}
498 instance Eq a => Eq (Set a) where
499 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
500
501 {--------------------------------------------------------------------
502 Ord
503 --------------------------------------------------------------------}
504
505 instance Ord a => Ord (Set a) where
506 compare s1 s2 = compare (toAscList s1) (toAscList s2)
507
508 {--------------------------------------------------------------------
509 Monoid
510 --------------------------------------------------------------------}
511
512 instance Ord a => Monoid (Set a) where
513 mempty = empty
514 mappend = union
515 mconcat = unions
516
517 {--------------------------------------------------------------------
518 Show
519 --------------------------------------------------------------------}
520 instance Show a => Show (Set a) where
521 showsPrec d s = showSet (toAscList s)
522
523 showSet :: (Show a) => [a] -> ShowS
524 showSet []
525 = showString "{}"
526 showSet (x:xs)
527 = showChar '{' . shows x . showTail xs
528 where
529 showTail [] = showChar '}'
530 showTail (x:xs) = showChar ',' . shows x . showTail xs
531
532
533 {--------------------------------------------------------------------
534 Typeable/Data
535 --------------------------------------------------------------------}
536
537 #include "Typeable.h"
538 INSTANCE_TYPEABLE1(Set,setTc,"Set")
539
540 {--------------------------------------------------------------------
541 Utility functions that return sub-ranges of the original
542 tree. Some functions take a comparison function as argument to
543 allow comparisons against infinite values. A function [cmplo x]
544 should be read as [compare lo x].
545
546 [trim cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
547 and [cmphi x == GT] for the value [x] of the root.
548 [filterGt cmp t] A tree where for all values [k]. [cmp k == LT]
549 [filterLt cmp t] A tree where for all values [k]. [cmp k == GT]
550
551 [split k t] Returns two trees [l] and [r] where all values
552 in [l] are <[k] and all keys in [r] are >[k].
553 [splitMember k t] Just like [split] but also returns whether [k]
554 was found in the tree.
555 --------------------------------------------------------------------}
556
557 {--------------------------------------------------------------------
558 [trim lo hi t] trims away all subtrees that surely contain no
559 values between the range [lo] to [hi]. The returned tree is either
560 empty or the key of the root is between @lo@ and @hi@.
561 --------------------------------------------------------------------}
562 trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a
563 trim cmplo cmphi Tip = Tip
564 trim cmplo cmphi t@(Bin sx x l r)
565 = case cmplo x of
566 LT -> case cmphi x of
567 GT -> t
568 le -> trim cmplo cmphi l
569 ge -> trim cmplo cmphi r
570
571 trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a)
572 trimMemberLo lo cmphi Tip = (False,Tip)
573 trimMemberLo lo cmphi t@(Bin sx x l r)
574 = case compare lo x of
575 LT -> case cmphi x of
576 GT -> (member lo t, t)
577 le -> trimMemberLo lo cmphi l
578 GT -> trimMemberLo lo cmphi r
579 EQ -> (True,trim (compare lo) cmphi r)
580
581
582 {--------------------------------------------------------------------
583 [filterGt x t] filter all values >[x] from tree [t]
584 [filterLt x t] filter all values <[x] from tree [t]
585 --------------------------------------------------------------------}
586 filterGt :: (a -> Ordering) -> Set a -> Set a
587 filterGt cmp Tip = Tip
588 filterGt cmp (Bin sx x l r)
589 = case cmp x of
590 LT -> join x (filterGt cmp l) r
591 GT -> filterGt cmp r
592 EQ -> r
593
594 filterLt :: (a -> Ordering) -> Set a -> Set a
595 filterLt cmp Tip = Tip
596 filterLt cmp (Bin sx x l r)
597 = case cmp x of
598 LT -> filterLt cmp l
599 GT -> join x l (filterLt cmp r)
600 EQ -> l
601
602
603 {--------------------------------------------------------------------
604 Split
605 --------------------------------------------------------------------}
606 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
607 -- where all elements in @set1@ are lower than @x@ and all elements in
608 -- @set2@ larger than @x@. @x@ is not found in neither @set1@ nor @set2@.
609 split :: Ord a => a -> Set a -> (Set a,Set a)
610 split x Tip = (Tip,Tip)
611 split x (Bin sy y l r)
612 = case compare x y of
613 LT -> let (lt,gt) = split x l in (lt,join y gt r)
614 GT -> let (lt,gt) = split x r in (join y l lt,gt)
615 EQ -> (l,r)
616
617 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
618 -- element was found in the original set.
619 splitMember :: Ord a => a -> Set a -> (Bool,Set a,Set a)
620 splitMember x Tip = (False,Tip,Tip)
621 splitMember x (Bin sy y l r)
622 = case compare x y of
623 LT -> let (found,lt,gt) = splitMember x l in (found,lt,join y gt r)
624 GT -> let (found,lt,gt) = splitMember x r in (found,join y l lt,gt)
625 EQ -> (True,l,r)
626
627 {--------------------------------------------------------------------
628 Utility functions that maintain the balance properties of the tree.
629 All constructors assume that all values in [l] < [x] and all values
630 in [r] > [x], and that [l] and [r] are valid trees.
631
632 In order of sophistication:
633 [Bin sz x l r] The type constructor.
634 [bin x l r] Maintains the correct size, assumes that both [l]
635 and [r] are balanced with respect to each other.
636 [balance x l r] Restores the balance and size.
637 Assumes that the original tree was balanced and
638 that [l] or [r] has changed by at most one element.
639 [join x l r] Restores balance and size.
640
641 Furthermore, we can construct a new tree from two trees. Both operations
642 assume that all values in [l] < all values in [r] and that [l] and [r]
643 are valid:
644 [glue l r] Glues [l] and [r] together. Assumes that [l] and
645 [r] are already balanced with respect to each other.
646 [merge l r] Merges two trees and restores balance.
647
648 Note: in contrast to Adam's paper, we use (<=) comparisons instead
649 of (<) comparisons in [join], [merge] and [balance].
650 Quickcheck (on [difference]) showed that this was necessary in order
651 to maintain the invariants. It is quite unsatisfactory that I haven't
652 been able to find out why this is actually the case! Fortunately, it
653 doesn't hurt to be a bit more conservative.
654 --------------------------------------------------------------------}
655
656 {--------------------------------------------------------------------
657 Join
658 --------------------------------------------------------------------}
659 join :: a -> Set a -> Set a -> Set a
660 join x Tip r = insertMin x r
661 join x l Tip = insertMax x l
662 join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
663 | delta*sizeL <= sizeR = balance z (join x l lz) rz
664 | delta*sizeR <= sizeL = balance y ly (join x ry r)
665 | otherwise = bin x l r
666
667
668 -- insertMin and insertMax don't perform potentially expensive comparisons.
669 insertMax,insertMin :: a -> Set a -> Set a
670 insertMax x t
671 = case t of
672 Tip -> singleton x
673 Bin sz y l r
674 -> balance y l (insertMax x r)
675
676 insertMin x t
677 = case t of
678 Tip -> singleton x
679 Bin sz y l r
680 -> balance y (insertMin x l) r
681
682 {--------------------------------------------------------------------
683 [merge l r]: merges two trees.
684 --------------------------------------------------------------------}
685 merge :: Set a -> Set a -> Set a
686 merge Tip r = r
687 merge l Tip = l
688 merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
689 | delta*sizeL <= sizeR = balance y (merge l ly) ry
690 | delta*sizeR <= sizeL = balance x lx (merge rx r)
691 | otherwise = glue l r
692
693 {--------------------------------------------------------------------
694 [glue l r]: glues two trees together.
695 Assumes that [l] and [r] are already balanced with respect to each other.
696 --------------------------------------------------------------------}
697 glue :: Set a -> Set a -> Set a
698 glue Tip r = r
699 glue l Tip = l
700 glue l r
701 | size l > size r = let (m,l') = deleteFindMax l in balance m l' r
702 | otherwise = let (m,r') = deleteFindMin r in balance m l r'
703
704
705 -- | /O(log n)/. Delete and find the minimal element.
706 --
707 -- > deleteFindMin set = (findMin set, deleteMin set)
708
709 deleteFindMin :: Set a -> (a,Set a)
710 deleteFindMin t
711 = case t of
712 Bin _ x Tip r -> (x,r)
713 Bin _ x l r -> let (xm,l') = deleteFindMin l in (xm,balance x l' r)
714 Tip -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)
715
716 -- | /O(log n)/. Delete and find the maximal element.
717 --
718 -- > deleteFindMax set = (findMax set, deleteMax set)
719 deleteFindMax :: Set a -> (a,Set a)
720 deleteFindMax t
721 = case t of
722 Bin _ x l Tip -> (x,l)
723 Bin _ x l r -> let (xm,r') = deleteFindMax r in (xm,balance x l r')
724 Tip -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)
725
726
727 {--------------------------------------------------------------------
728 [balance x l r] balances two trees with value x.
729 The sizes of the trees should balance after decreasing the
730 size of one of them. (a rotation).
731
732 [delta] is the maximal relative difference between the sizes of
733 two trees, it corresponds with the [w] in Adams' paper,
734 or equivalently, [1/delta] corresponds with the $\alpha$
735 in Nievergelt's paper. Adams shows that [delta] should
736 be larger than 3.745 in order to garantee that the
737 rotations can always restore balance.
738
739 [ratio] is the ratio between an outer and inner sibling of the
740 heavier subtree in an unbalanced setting. It determines
741 whether a double or single rotation should be performed
742 to restore balance. It is correspondes with the inverse
743 of $\alpha$ in Adam's article.
744
745 Note that:
746 - [delta] should be larger than 4.646 with a [ratio] of 2.
747 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
748
749 - A lower [delta] leads to a more 'perfectly' balanced tree.
750 - A higher [delta] performs less rebalancing.
751
752 - Balancing is automatic for random data and a balancing
753 scheme is only necessary to avoid pathological worst cases.
754 Almost any choice will do in practice
755
756 - Allthough it seems that a rather large [delta] may perform better
757 than smaller one, measurements have shown that the smallest [delta]
758 of 4 is actually the fastest on a wide range of operations. It
759 especially improves performance on worst-case scenarios like
760 a sequence of ordered insertions.
761
762 Note: in contrast to Adams' paper, we use a ratio of (at least) 2
763 to decide whether a single or double rotation is needed. Allthough
764 he actually proves that this ratio is needed to maintain the
765 invariants, his implementation uses a (invalid) ratio of 1.
766 He is aware of the problem though since he has put a comment in his
767 original source code that he doesn't care about generating a
768 slightly inbalanced tree since it doesn't seem to matter in practice.
769 However (since we use quickcheck :-) we will stick to strictly balanced
770 trees.
771 --------------------------------------------------------------------}
772 delta,ratio :: Int
773 delta = 4
774 ratio = 2
775
776 balance :: a -> Set a -> Set a -> Set a
777 balance x l r
778 | sizeL + sizeR <= 1 = Bin sizeX x l r
779 | sizeR >= delta*sizeL = rotateL x l r
780 | sizeL >= delta*sizeR = rotateR x l r
781 | otherwise = Bin sizeX x l r
782 where
783 sizeL = size l
784 sizeR = size r
785 sizeX = sizeL + sizeR + 1
786
787 -- rotate
788 rotateL x l r@(Bin _ _ ly ry)
789 | size ly < ratio*size ry = singleL x l r
790 | otherwise = doubleL x l r
791
792 rotateR x l@(Bin _ _ ly ry) r
793 | size ry < ratio*size ly = singleR x l r
794 | otherwise = doubleR x l r
795
796 -- basic rotations
797 singleL x1 t1 (Bin _ x2 t2 t3) = bin x2 (bin x1 t1 t2) t3
798 singleR x1 (Bin _ x2 t1 t2) t3 = bin x2 t1 (bin x1 t2 t3)
799
800 doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
801 doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)
802
803
804 {--------------------------------------------------------------------
805 The bin constructor maintains the size of the tree
806 --------------------------------------------------------------------}
807 bin :: a -> Set a -> Set a -> Set a
808 bin x l r
809 = Bin (size l + size r + 1) x l r
810
811
812 {--------------------------------------------------------------------
813 Utilities
814 --------------------------------------------------------------------}
815 foldlStrict f z xs
816 = case xs of
817 [] -> z
818 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
819
820
821 {--------------------------------------------------------------------
822 Debugging
823 --------------------------------------------------------------------}
824 -- | /O(n)/. Show the tree that implements the set. The tree is shown
825 -- in a compressed, hanging format.
826 showTree :: Show a => Set a -> String
827 showTree s
828 = showTreeWith True False s
829
830
831 {- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
832 the tree that implements the set. If @hang@ is
833 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
834 @wide@ is 'True', an extra wide version is shown.
835
836 > Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
837 > 4
838 > +--2
839 > | +--1
840 > | +--3
841 > +--5
842 >
843 > Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
844 > 4
845 > |
846 > +--2
847 > | |
848 > | +--1
849 > | |
850 > | +--3
851 > |
852 > +--5
853 >
854 > Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
855 > +--5
856 > |
857 > 4
858 > |
859 > | +--3
860 > | |
861 > +--2
862 > |
863 > +--1
864
865 -}
866 showTreeWith :: Show a => Bool -> Bool -> Set a -> String
867 showTreeWith hang wide t
868 | hang = (showsTreeHang wide [] t) ""
869 | otherwise = (showsTree wide [] [] t) ""
870
871 showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
872 showsTree wide lbars rbars t
873 = case t of
874 Tip -> showsBars lbars . showString "|\n"
875 Bin sz x Tip Tip
876 -> showsBars lbars . shows x . showString "\n"
877 Bin sz x l r
878 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
879 showWide wide rbars .
880 showsBars lbars . shows x . showString "\n" .
881 showWide wide lbars .
882 showsTree wide (withEmpty lbars) (withBar lbars) l
883
884 showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
885 showsTreeHang wide bars t
886 = case t of
887 Tip -> showsBars bars . showString "|\n"
888 Bin sz x Tip Tip
889 -> showsBars bars . shows x . showString "\n"
890 Bin sz x l r
891 -> showsBars bars . shows x . showString "\n" .
892 showWide wide bars .
893 showsTreeHang wide (withBar bars) l .
894 showWide wide bars .
895 showsTreeHang wide (withEmpty bars) r
896
897
898 showWide wide bars
899 | wide = showString (concat (reverse bars)) . showString "|\n"
900 | otherwise = id
901
902 showsBars :: [String] -> ShowS
903 showsBars bars
904 = case bars of
905 [] -> id
906 _ -> showString (concat (reverse (tail bars))) . showString node
907
908 node = "+--"
909 withBar bars = "| ":bars
910 withEmpty bars = " ":bars
911
912 {--------------------------------------------------------------------
913 Assertions
914 --------------------------------------------------------------------}
915 -- | /O(n)/. Test if the internal set structure is valid.
916 valid :: Ord a => Set a -> Bool
917 valid t
918 = balanced t && ordered t && validsize t
919
920 ordered t
921 = bounded (const True) (const True) t
922 where
923 bounded lo hi t
924 = case t of
925 Tip -> True
926 Bin sz x l r -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r
927
928 balanced :: Set a -> Bool
929 balanced t
930 = case t of
931 Tip -> True
932 Bin sz x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
933 balanced l && balanced r
934
935
936 validsize t
937 = (realsize t == Just (size t))
938 where
939 realsize t
940 = case t of
941 Tip -> Just 0
942 Bin sz x l r -> case (realsize l,realsize r) of
943 (Just n,Just m) | n+m+1 == sz -> Just sz
944 other -> Nothing
945
946 {-
947 {--------------------------------------------------------------------
948 Testing
949 --------------------------------------------------------------------}
950 testTree :: [Int] -> Set Int
951 testTree xs = fromList xs
952 test1 = testTree [1..20]
953 test2 = testTree [30,29..10]
954 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
955
956 {--------------------------------------------------------------------
957 QuickCheck
958 --------------------------------------------------------------------}
959 qcheck prop
960 = check config prop
961 where
962 config = Config
963 { configMaxTest = 500
964 , configMaxFail = 5000
965 , configSize = \n -> (div n 2 + 3)
966 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
967 }
968
969
970 {--------------------------------------------------------------------
971 Arbitrary, reasonably balanced trees
972 --------------------------------------------------------------------}
973 instance (Enum a) => Arbitrary (Set a) where
974 arbitrary = sized (arbtree 0 maxkey)
975 where maxkey = 10000
976
977 arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a)
978 arbtree lo hi n
979 | n <= 0 = return Tip
980 | lo >= hi = return Tip
981 | otherwise = do{ i <- choose (lo,hi)
982 ; m <- choose (1,30)
983 ; let (ml,mr) | m==(1::Int)= (1,2)
984 | m==2 = (2,1)
985 | m==3 = (1,1)
986 | otherwise = (2,2)
987 ; l <- arbtree lo (i-1) (n `div` ml)
988 ; r <- arbtree (i+1) hi (n `div` mr)
989 ; return (bin (toEnum i) l r)
990 }
991
992
993 {--------------------------------------------------------------------
994 Valid tree's
995 --------------------------------------------------------------------}
996 forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property
997 forValid f
998 = forAll arbitrary $ \t ->
999 -- classify (balanced t) "balanced" $
1000 classify (size t == 0) "empty" $
1001 classify (size t > 0 && size t <= 10) "small" $
1002 classify (size t > 10 && size t <= 64) "medium" $
1003 classify (size t > 64) "large" $
1004 balanced t ==> f t
1005
1006 forValidIntTree :: Testable a => (Set Int -> a) -> Property
1007 forValidIntTree f
1008 = forValid f
1009
1010 forValidUnitTree :: Testable a => (Set Int -> a) -> Property
1011 forValidUnitTree f
1012 = forValid f
1013
1014
1015 prop_Valid
1016 = forValidUnitTree $ \t -> valid t
1017
1018 {--------------------------------------------------------------------
1019 Single, Insert, Delete
1020 --------------------------------------------------------------------}
1021 prop_Single :: Int -> Bool
1022 prop_Single x
1023 = (insert x empty == singleton x)
1024
1025 prop_InsertValid :: Int -> Property
1026 prop_InsertValid k
1027 = forValidUnitTree $ \t -> valid (insert k t)
1028
1029 prop_InsertDelete :: Int -> Set Int -> Property
1030 prop_InsertDelete k t
1031 = not (member k t) ==> delete k (insert k t) == t
1032
1033 prop_DeleteValid :: Int -> Property
1034 prop_DeleteValid k
1035 = forValidUnitTree $ \t ->
1036 valid (delete k (insert k t))
1037
1038 {--------------------------------------------------------------------
1039 Balance
1040 --------------------------------------------------------------------}
1041 prop_Join :: Int -> Property
1042 prop_Join x
1043 = forValidUnitTree $ \t ->
1044 let (l,r) = split x t
1045 in valid (join x l r)
1046
1047 prop_Merge :: Int -> Property
1048 prop_Merge x
1049 = forValidUnitTree $ \t ->
1050 let (l,r) = split x t
1051 in valid (merge l r)
1052
1053
1054 {--------------------------------------------------------------------
1055 Union
1056 --------------------------------------------------------------------}
1057 prop_UnionValid :: Property
1058 prop_UnionValid
1059 = forValidUnitTree $ \t1 ->
1060 forValidUnitTree $ \t2 ->
1061 valid (union t1 t2)
1062
1063 prop_UnionInsert :: Int -> Set Int -> Bool
1064 prop_UnionInsert x t
1065 = union t (singleton x) == insert x t
1066
1067 prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool
1068 prop_UnionAssoc t1 t2 t3
1069 = union t1 (union t2 t3) == union (union t1 t2) t3
1070
1071 prop_UnionComm :: Set Int -> Set Int -> Bool
1072 prop_UnionComm t1 t2
1073 = (union t1 t2 == union t2 t1)
1074
1075
1076 prop_DiffValid
1077 = forValidUnitTree $ \t1 ->
1078 forValidUnitTree $ \t2 ->
1079 valid (difference t1 t2)
1080
1081 prop_Diff :: [Int] -> [Int] -> Bool
1082 prop_Diff xs ys
1083 = toAscList (difference (fromList xs) (fromList ys))
1084 == List.sort ((List.\\) (nub xs) (nub ys))
1085
1086 prop_IntValid
1087 = forValidUnitTree $ \t1 ->
1088 forValidUnitTree $ \t2 ->
1089 valid (intersection t1 t2)
1090
1091 prop_Int :: [Int] -> [Int] -> Bool
1092 prop_Int xs ys
1093 = toAscList (intersection (fromList xs) (fromList ys))
1094 == List.sort (nub ((List.intersect) (xs) (ys)))
1095
1096 {--------------------------------------------------------------------
1097 Lists
1098 --------------------------------------------------------------------}
1099 prop_Ordered
1100 = forAll (choose (5,100)) $ \n ->
1101 let xs = [0..n::Int]
1102 in fromAscList xs == fromList xs
1103
1104 prop_List :: [Int] -> Bool
1105 prop_List xs
1106 = (sort (nub xs) == toList (fromList xs))
1107 -}
1108
1109 {--------------------------------------------------------------------
1110 Old Data.Set compatibility interface
1111 --------------------------------------------------------------------}
1112
1113 {-# DEPRECATED emptySet "Use empty instead" #-}
1114 -- | Obsolete equivalent of 'empty'.
1115 emptySet :: Set a
1116 emptySet = empty
1117
1118 {-# DEPRECATED mkSet "Use fromList instead" #-}
1119 -- | Obsolete equivalent of 'fromList'.
1120 mkSet :: Ord a => [a] -> Set a
1121 mkSet = fromList
1122
1123 {-# DEPRECATED setToList "Use elems instead." #-}
1124 -- | Obsolete equivalent of 'elems'.
1125 setToList :: Set a -> [a]
1126 setToList = elems
1127
1128 {-# DEPRECATED unitSet "Use singleton instead." #-}
1129 -- | Obsolete equivalent of 'singleton'.
1130 unitSet :: a -> Set a
1131 unitSet = singleton
1132
1133 {-# DEPRECATED elementOf "Use member instead." #-}
1134 -- | Obsolete equivalent of 'member'.
1135 elementOf :: Ord a => a -> Set a -> Bool
1136 elementOf = member
1137
1138 {-# DEPRECATED isEmptySet "Use null instead." #-}
1139 -- | Obsolete equivalent of 'null'.
1140 isEmptySet :: Set a -> Bool
1141 isEmptySet = null
1142
1143 {-# DEPRECATED cardinality "Use size instead." #-}
1144 -- | Obsolete equivalent of 'size'.
1145 cardinality :: Set a -> Int
1146 cardinality = size
1147
1148 {-# DEPRECATED unionManySets "Use unions instead." #-}
1149 -- | Obsolete equivalent of 'unions'.
1150 unionManySets :: Ord a => [Set a] -> Set a
1151 unionManySets = unions
1152
1153 {-# DEPRECATED minusSet "Use difference instead." #-}
1154 -- | Obsolete equivalent of 'difference'.
1155 minusSet :: Ord a => Set a -> Set a -> Set a
1156 minusSet = difference
1157
1158 {-# DEPRECATED mapSet "Use map instead." #-}
1159 -- | Obsolete equivalent of 'map'.
1160 mapSet :: (Ord a, Ord b) => (b -> a) -> Set b -> Set a
1161 mapSet = map
1162
1163 {-# DEPRECATED intersect "Use intersection instead." #-}
1164 -- | Obsolete equivalent of 'intersection'.
1165 intersect :: Ord a => Set a -> Set a -> Set a
1166 intersect = intersection
1167
1168 {-# DEPRECATED addToSet "Use 'flip insert' instead." #-}
1169 -- | Obsolete equivalent of @'flip' 'insert'@.
1170 addToSet :: Ord a => Set a -> a -> Set a
1171 addToSet = flip insert
1172
1173 {-# DEPRECATED delFromSet "Use `flip delete' instead." #-}
1174 -- | Obsolete equivalent of @'flip' 'delete'@.
1175 delFromSet :: Ord a => Set a -> a -> Set a
1176 delFromSet = flip delete