expand advice on importing these modules
[packages/random.git] / Data / IntSet.hs
1 {-# OPTIONS -cpp -fglasgow-exts #-}
2 -----------------------------------------------------------------------------
3 -- |
4 -- Module : Data.IntSet
5 -- Copyright : (c) Daan Leijen 2002
6 -- License : BSD-style
7 -- Maintainer : libraries@haskell.org
8 -- Stability : provisional
9 -- Portability : portable
10 --
11 -- An efficient implementation of integer sets.
12 --
13 -- Since many function names (but not the type name) clash with
14 -- "Prelude" names, this module is usually imported @qualified@, e.g.
15 --
16 -- > import Data.IntSet (IntSet)
17 -- > import qualified Data.IntSet as IntSet
18 --
19 -- The implementation is based on /big-endian patricia trees/. This data
20 -- structure performs especially well on binary operations like 'union'
21 -- and 'intersection'. However, my benchmarks show that it is also
22 -- (much) faster on insertions and deletions when compared to a generic
23 -- size-balanced set implementation (see "Data.Set").
24 --
25 -- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
26 -- Workshop on ML, September 1998, pages 77-86,
27 -- <http://www.cse.ogi.edu/~andy/pub/finite.htm>
28 --
29 -- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
30 -- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
31 -- October 1968, pages 514-534.
32 --
33 -- Many operations have a worst-case complexity of /O(min(n,W))/.
34 -- This means that the operation can become linear in the number of
35 -- elements with a maximum of /W/ -- the number of bits in an 'Int'
36 -- (32 or 64).
37 -----------------------------------------------------------------------------
38
39 module Data.IntSet (
40 -- * Set type
41 IntSet -- instance Eq,Show
42
43 -- * Operators
44 , (\\)
45
46 -- * Query
47 , null
48 , size
49 , member
50 , notMember
51 , isSubsetOf
52 , isProperSubsetOf
53
54 -- * Construction
55 , empty
56 , singleton
57 , insert
58 , delete
59
60 -- * Combine
61 , union, unions
62 , difference
63 , intersection
64
65 -- * Filter
66 , filter
67 , partition
68 , split
69 , splitMember
70
71 -- * Map
72 , map
73
74 -- * Fold
75 , fold
76
77 -- * Conversion
78 -- ** List
79 , elems
80 , toList
81 , fromList
82
83 -- ** Ordered list
84 , toAscList
85 , fromAscList
86 , fromDistinctAscList
87
88 -- * Debugging
89 , showTree
90 , showTreeWith
91 ) where
92
93
94 import Prelude hiding (lookup,filter,foldr,foldl,null,map)
95 import Data.Bits
96 import Data.Int
97
98 import qualified Data.List as List
99 import Data.Monoid (Monoid(..))
100 import Data.Typeable
101
102 {-
103 -- just for testing
104 import QuickCheck
105 import List (nub,sort)
106 import qualified List
107 -}
108
109 #if __GLASGOW_HASKELL__
110 import Text.Read
111 import Data.Generics.Basics
112 import Data.Generics.Instances
113 #endif
114
115 #if __GLASGOW_HASKELL__ >= 503
116 import GHC.Word
117 import GHC.Exts ( Word(..), Int(..), shiftRL# )
118 #elif __GLASGOW_HASKELL__
119 import Word
120 import GlaExts ( Word(..), Int(..), shiftRL# )
121 #else
122 import Data.Word
123 #endif
124
125 infixl 9 \\{-This comment teaches CPP correct behaviour -}
126
127 -- A "Nat" is a natural machine word (an unsigned Int)
128 type Nat = Word
129
130 natFromInt :: Int -> Nat
131 natFromInt i = fromIntegral i
132
133 intFromNat :: Nat -> Int
134 intFromNat w = fromIntegral w
135
136 shiftRL :: Nat -> Int -> Nat
137 #if __GLASGOW_HASKELL__
138 {--------------------------------------------------------------------
139 GHC: use unboxing to get @shiftRL@ inlined.
140 --------------------------------------------------------------------}
141 shiftRL (W# x) (I# i)
142 = W# (shiftRL# x i)
143 #else
144 shiftRL x i = shiftR x i
145 #endif
146
147 {--------------------------------------------------------------------
148 Operators
149 --------------------------------------------------------------------}
150 -- | /O(n+m)/. See 'difference'.
151 (\\) :: IntSet -> IntSet -> IntSet
152 m1 \\ m2 = difference m1 m2
153
154 {--------------------------------------------------------------------
155 Types
156 --------------------------------------------------------------------}
157 -- | A set of integers.
158 data IntSet = Nil
159 | Tip {-# UNPACK #-} !Int
160 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
161
162 type Prefix = Int
163 type Mask = Int
164
165 instance Monoid IntSet where
166 mempty = empty
167 mappend = union
168 mconcat = unions
169
170 #if __GLASGOW_HASKELL__
171
172 {--------------------------------------------------------------------
173 A Data instance
174 --------------------------------------------------------------------}
175
176 -- This instance preserves data abstraction at the cost of inefficiency.
177 -- We omit reflection services for the sake of data abstraction.
178
179 instance Data IntSet where
180 gfoldl f z is = z fromList `f` (toList is)
181 toConstr _ = error "toConstr"
182 gunfold _ _ = error "gunfold"
183 dataTypeOf _ = mkNorepType "Data.IntSet.IntSet"
184
185 #endif
186
187 {--------------------------------------------------------------------
188 Query
189 --------------------------------------------------------------------}
190 -- | /O(1)/. Is the set empty?
191 null :: IntSet -> Bool
192 null Nil = True
193 null other = False
194
195 -- | /O(n)/. Cardinality of the set.
196 size :: IntSet -> Int
197 size t
198 = case t of
199 Bin p m l r -> size l + size r
200 Tip y -> 1
201 Nil -> 0
202
203 -- | /O(min(n,W))/. Is the value a member of the set?
204 member :: Int -> IntSet -> Bool
205 member x t
206 = case t of
207 Bin p m l r
208 | nomatch x p m -> False
209 | zero x m -> member x l
210 | otherwise -> member x r
211 Tip y -> (x==y)
212 Nil -> False
213
214 -- | /O(log n)/. Is the element not in the set?
215 notMember :: Int -> IntSet -> Bool
216 notMember k = not . member k
217
218 -- 'lookup' is used by 'intersection' for left-biasing
219 lookup :: Int -> IntSet -> Maybe Int
220 lookup k t
221 = let nk = natFromInt k in seq nk (lookupN nk t)
222
223 lookupN :: Nat -> IntSet -> Maybe Int
224 lookupN k t
225 = case t of
226 Bin p m l r
227 | zeroN k (natFromInt m) -> lookupN k l
228 | otherwise -> lookupN k r
229 Tip kx
230 | (k == natFromInt kx) -> Just kx
231 | otherwise -> Nothing
232 Nil -> Nothing
233
234 {--------------------------------------------------------------------
235 Construction
236 --------------------------------------------------------------------}
237 -- | /O(1)/. The empty set.
238 empty :: IntSet
239 empty
240 = Nil
241
242 -- | /O(1)/. A set of one element.
243 singleton :: Int -> IntSet
244 singleton x
245 = Tip x
246
247 {--------------------------------------------------------------------
248 Insert
249 --------------------------------------------------------------------}
250 -- | /O(min(n,W))/. Add a value to the set. When the value is already
251 -- an element of the set, it is replaced by the new one, ie. 'insert'
252 -- is left-biased.
253 insert :: Int -> IntSet -> IntSet
254 insert x t
255 = case t of
256 Bin p m l r
257 | nomatch x p m -> join x (Tip x) p t
258 | zero x m -> Bin p m (insert x l) r
259 | otherwise -> Bin p m l (insert x r)
260 Tip y
261 | x==y -> Tip x
262 | otherwise -> join x (Tip x) y t
263 Nil -> Tip x
264
265 -- right-biased insertion, used by 'union'
266 insertR :: Int -> IntSet -> IntSet
267 insertR x t
268 = case t of
269 Bin p m l r
270 | nomatch x p m -> join x (Tip x) p t
271 | zero x m -> Bin p m (insert x l) r
272 | otherwise -> Bin p m l (insert x r)
273 Tip y
274 | x==y -> t
275 | otherwise -> join x (Tip x) y t
276 Nil -> Tip x
277
278 -- | /O(min(n,W))/. Delete a value in the set. Returns the
279 -- original set when the value was not present.
280 delete :: Int -> IntSet -> IntSet
281 delete x t
282 = case t of
283 Bin p m l r
284 | nomatch x p m -> t
285 | zero x m -> bin p m (delete x l) r
286 | otherwise -> bin p m l (delete x r)
287 Tip y
288 | x==y -> Nil
289 | otherwise -> t
290 Nil -> Nil
291
292
293 {--------------------------------------------------------------------
294 Union
295 --------------------------------------------------------------------}
296 -- | The union of a list of sets.
297 unions :: [IntSet] -> IntSet
298 unions xs
299 = foldlStrict union empty xs
300
301
302 -- | /O(n+m)/. The union of two sets.
303 union :: IntSet -> IntSet -> IntSet
304 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
305 | shorter m1 m2 = union1
306 | shorter m2 m1 = union2
307 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
308 | otherwise = join p1 t1 p2 t2
309 where
310 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
311 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
312 | otherwise = Bin p1 m1 l1 (union r1 t2)
313
314 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
315 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
316 | otherwise = Bin p2 m2 l2 (union t1 r2)
317
318 union (Tip x) t = insert x t
319 union t (Tip x) = insertR x t -- right bias
320 union Nil t = t
321 union t Nil = t
322
323
324 {--------------------------------------------------------------------
325 Difference
326 --------------------------------------------------------------------}
327 -- | /O(n+m)/. Difference between two sets.
328 difference :: IntSet -> IntSet -> IntSet
329 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
330 | shorter m1 m2 = difference1
331 | shorter m2 m1 = difference2
332 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
333 | otherwise = t1
334 where
335 difference1 | nomatch p2 p1 m1 = t1
336 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
337 | otherwise = bin p1 m1 l1 (difference r1 t2)
338
339 difference2 | nomatch p1 p2 m2 = t1
340 | zero p1 m2 = difference t1 l2
341 | otherwise = difference t1 r2
342
343 difference t1@(Tip x) t2
344 | member x t2 = Nil
345 | otherwise = t1
346
347 difference Nil t = Nil
348 difference t (Tip x) = delete x t
349 difference t Nil = t
350
351
352
353 {--------------------------------------------------------------------
354 Intersection
355 --------------------------------------------------------------------}
356 -- | /O(n+m)/. The intersection of two sets.
357 intersection :: IntSet -> IntSet -> IntSet
358 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
359 | shorter m1 m2 = intersection1
360 | shorter m2 m1 = intersection2
361 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
362 | otherwise = Nil
363 where
364 intersection1 | nomatch p2 p1 m1 = Nil
365 | zero p2 m1 = intersection l1 t2
366 | otherwise = intersection r1 t2
367
368 intersection2 | nomatch p1 p2 m2 = Nil
369 | zero p1 m2 = intersection t1 l2
370 | otherwise = intersection t1 r2
371
372 intersection t1@(Tip x) t2
373 | member x t2 = t1
374 | otherwise = Nil
375 intersection t (Tip x)
376 = case lookup x t of
377 Just y -> Tip y
378 Nothing -> Nil
379 intersection Nil t = Nil
380 intersection t Nil = Nil
381
382
383
384 {--------------------------------------------------------------------
385 Subset
386 --------------------------------------------------------------------}
387 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
388 isProperSubsetOf :: IntSet -> IntSet -> Bool
389 isProperSubsetOf t1 t2
390 = case subsetCmp t1 t2 of
391 LT -> True
392 ge -> False
393
394 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
395 | shorter m1 m2 = GT
396 | shorter m2 m1 = subsetCmpLt
397 | p1 == p2 = subsetCmpEq
398 | otherwise = GT -- disjoint
399 where
400 subsetCmpLt | nomatch p1 p2 m2 = GT
401 | zero p1 m2 = subsetCmp t1 l2
402 | otherwise = subsetCmp t1 r2
403 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
404 (GT,_ ) -> GT
405 (_ ,GT) -> GT
406 (EQ,EQ) -> EQ
407 other -> LT
408
409 subsetCmp (Bin p m l r) t = GT
410 subsetCmp (Tip x) (Tip y)
411 | x==y = EQ
412 | otherwise = GT -- disjoint
413 subsetCmp (Tip x) t
414 | member x t = LT
415 | otherwise = GT -- disjoint
416 subsetCmp Nil Nil = EQ
417 subsetCmp Nil t = LT
418
419 -- | /O(n+m)/. Is this a subset?
420 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
421
422 isSubsetOf :: IntSet -> IntSet -> Bool
423 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
424 | shorter m1 m2 = False
425 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
426 else isSubsetOf t1 r2)
427 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
428 isSubsetOf (Bin p m l r) t = False
429 isSubsetOf (Tip x) t = member x t
430 isSubsetOf Nil t = True
431
432
433 {--------------------------------------------------------------------
434 Filter
435 --------------------------------------------------------------------}
436 -- | /O(n)/. Filter all elements that satisfy some predicate.
437 filter :: (Int -> Bool) -> IntSet -> IntSet
438 filter pred t
439 = case t of
440 Bin p m l r
441 -> bin p m (filter pred l) (filter pred r)
442 Tip x
443 | pred x -> t
444 | otherwise -> Nil
445 Nil -> Nil
446
447 -- | /O(n)/. partition the set according to some predicate.
448 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
449 partition pred t
450 = case t of
451 Bin p m l r
452 -> let (l1,l2) = partition pred l
453 (r1,r2) = partition pred r
454 in (bin p m l1 r1, bin p m l2 r2)
455 Tip x
456 | pred x -> (t,Nil)
457 | otherwise -> (Nil,t)
458 Nil -> (Nil,Nil)
459
460
461 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
462 -- where all elements in @set1@ are lower than @x@ and all elements in
463 -- @set2@ larger than @x@.
464 --
465 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
466 split :: Int -> IntSet -> (IntSet,IntSet)
467 split x t
468 = case t of
469 Bin p m l r
470 | m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
471 else let (lt,gt) = split' x r in (lt, union gt l)
472 -- handle negative numbers.
473 | otherwise -> split' x t
474 Tip y
475 | x>y -> (t,Nil)
476 | x<y -> (Nil,t)
477 | otherwise -> (Nil,Nil)
478 Nil -> (Nil, Nil)
479
480 split' :: Int -> IntSet -> (IntSet,IntSet)
481 split' x t
482 = case t of
483 Bin p m l r
484 | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
485 else let (lt,gt) = split' x r in (union l lt,gt)
486 | otherwise -> if x < p then (Nil, t)
487 else (t, Nil)
488 Tip y
489 | x>y -> (t,Nil)
490 | x<y -> (Nil,t)
491 | otherwise -> (Nil,Nil)
492 Nil -> (Nil,Nil)
493
494 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
495 -- element was found in the original set.
496 splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
497 splitMember x t
498 = case t of
499 Bin p m l r
500 | m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
501 else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
502 -- handle negative numbers.
503 | otherwise -> splitMember' x t
504 Tip y
505 | x>y -> (t,False,Nil)
506 | x<y -> (Nil,False,t)
507 | otherwise -> (Nil,True,Nil)
508 Nil -> (Nil,False,Nil)
509
510 splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
511 splitMember' x t
512 = case t of
513 Bin p m l r
514 | match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
515 else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
516 | otherwise -> if x < p then (Nil, False, t)
517 else (t, False, Nil)
518 Tip y
519 | x>y -> (t,False,Nil)
520 | x<y -> (Nil,False,t)
521 | otherwise -> (Nil,True,Nil)
522 Nil -> (Nil,False,Nil)
523
524 {----------------------------------------------------------------------
525 Map
526 ----------------------------------------------------------------------}
527
528 -- | /O(n*min(n,W))/.
529 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
530 --
531 -- It's worth noting that the size of the result may be smaller if,
532 -- for some @(x,y)@, @x \/= y && f x == f y@
533
534 map :: (Int->Int) -> IntSet -> IntSet
535 map f = fromList . List.map f . toList
536
537 {--------------------------------------------------------------------
538 Fold
539 --------------------------------------------------------------------}
540 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
541 --
542 -- > sum set == fold (+) 0 set
543 -- > elems set == fold (:) [] set
544 fold :: (Int -> b -> b) -> b -> IntSet -> b
545 fold f z t
546 = case t of
547 Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r
548 -- put negative numbers before.
549 Bin p m l r -> foldr f z t
550 Tip x -> f x z
551 Nil -> z
552
553 foldr :: (Int -> b -> b) -> b -> IntSet -> b
554 foldr f z t
555 = case t of
556 Bin p m l r -> foldr f (foldr f z r) l
557 Tip x -> f x z
558 Nil -> z
559
560 {--------------------------------------------------------------------
561 List variations
562 --------------------------------------------------------------------}
563 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
564 elems :: IntSet -> [Int]
565 elems s
566 = toList s
567
568 {--------------------------------------------------------------------
569 Lists
570 --------------------------------------------------------------------}
571 -- | /O(n)/. Convert the set to a list of elements.
572 toList :: IntSet -> [Int]
573 toList t
574 = fold (:) [] t
575
576 -- | /O(n)/. Convert the set to an ascending list of elements.
577 toAscList :: IntSet -> [Int]
578 toAscList t = toList t
579
580 -- | /O(n*min(n,W))/. Create a set from a list of integers.
581 fromList :: [Int] -> IntSet
582 fromList xs
583 = foldlStrict ins empty xs
584 where
585 ins t x = insert x t
586
587 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
588 fromAscList :: [Int] -> IntSet
589 fromAscList xs
590 = fromList xs
591
592 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
593 fromDistinctAscList :: [Int] -> IntSet
594 fromDistinctAscList xs
595 = fromList xs
596
597
598 {--------------------------------------------------------------------
599 Eq
600 --------------------------------------------------------------------}
601 instance Eq IntSet where
602 t1 == t2 = equal t1 t2
603 t1 /= t2 = nequal t1 t2
604
605 equal :: IntSet -> IntSet -> Bool
606 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
607 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
608 equal (Tip x) (Tip y)
609 = (x==y)
610 equal Nil Nil = True
611 equal t1 t2 = False
612
613 nequal :: IntSet -> IntSet -> Bool
614 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
615 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
616 nequal (Tip x) (Tip y)
617 = (x/=y)
618 nequal Nil Nil = False
619 nequal t1 t2 = True
620
621 {--------------------------------------------------------------------
622 Ord
623 --------------------------------------------------------------------}
624
625 instance Ord IntSet where
626 compare s1 s2 = compare (toAscList s1) (toAscList s2)
627 -- tentative implementation. See if more efficient exists.
628
629 {--------------------------------------------------------------------
630 Show
631 --------------------------------------------------------------------}
632 instance Show IntSet where
633 showsPrec p xs = showParen (p > 10) $
634 showString "fromList " . shows (toList xs)
635
636 showSet :: [Int] -> ShowS
637 showSet []
638 = showString "{}"
639 showSet (x:xs)
640 = showChar '{' . shows x . showTail xs
641 where
642 showTail [] = showChar '}'
643 showTail (x:xs) = showChar ',' . shows x . showTail xs
644
645 {--------------------------------------------------------------------
646 Read
647 --------------------------------------------------------------------}
648 instance Read IntSet where
649 #ifdef __GLASGOW_HASKELL__
650 readPrec = parens $ prec 10 $ do
651 Ident "fromList" <- lexP
652 xs <- readPrec
653 return (fromList xs)
654
655 readListPrec = readListPrecDefault
656 #else
657 readsPrec p = readParen (p > 10) $ \ r -> do
658 ("fromList",s) <- lex r
659 (xs,t) <- reads s
660 return (fromList xs,t)
661 #endif
662
663 {--------------------------------------------------------------------
664 Typeable
665 --------------------------------------------------------------------}
666
667 #include "Typeable.h"
668 INSTANCE_TYPEABLE0(IntSet,intSetTc,"IntSet")
669
670 {--------------------------------------------------------------------
671 Debugging
672 --------------------------------------------------------------------}
673 -- | /O(n)/. Show the tree that implements the set. The tree is shown
674 -- in a compressed, hanging format.
675 showTree :: IntSet -> String
676 showTree s
677 = showTreeWith True False s
678
679
680 {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
681 the tree that implements the set. If @hang@ is
682 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
683 @wide@ is 'True', an extra wide version is shown.
684 -}
685 showTreeWith :: Bool -> Bool -> IntSet -> String
686 showTreeWith hang wide t
687 | hang = (showsTreeHang wide [] t) ""
688 | otherwise = (showsTree wide [] [] t) ""
689
690 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
691 showsTree wide lbars rbars t
692 = case t of
693 Bin p m l r
694 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
695 showWide wide rbars .
696 showsBars lbars . showString (showBin p m) . showString "\n" .
697 showWide wide lbars .
698 showsTree wide (withEmpty lbars) (withBar lbars) l
699 Tip x
700 -> showsBars lbars . showString " " . shows x . showString "\n"
701 Nil -> showsBars lbars . showString "|\n"
702
703 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
704 showsTreeHang wide bars t
705 = case t of
706 Bin p m l r
707 -> showsBars bars . showString (showBin p m) . showString "\n" .
708 showWide wide bars .
709 showsTreeHang wide (withBar bars) l .
710 showWide wide bars .
711 showsTreeHang wide (withEmpty bars) r
712 Tip x
713 -> showsBars bars . showString " " . shows x . showString "\n"
714 Nil -> showsBars bars . showString "|\n"
715
716 showBin p m
717 = "*" -- ++ show (p,m)
718
719 showWide wide bars
720 | wide = showString (concat (reverse bars)) . showString "|\n"
721 | otherwise = id
722
723 showsBars :: [String] -> ShowS
724 showsBars bars
725 = case bars of
726 [] -> id
727 _ -> showString (concat (reverse (tail bars))) . showString node
728
729 node = "+--"
730 withBar bars = "| ":bars
731 withEmpty bars = " ":bars
732
733
734 {--------------------------------------------------------------------
735 Helpers
736 --------------------------------------------------------------------}
737 {--------------------------------------------------------------------
738 Join
739 --------------------------------------------------------------------}
740 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
741 join p1 t1 p2 t2
742 | zero p1 m = Bin p m t1 t2
743 | otherwise = Bin p m t2 t1
744 where
745 m = branchMask p1 p2
746 p = mask p1 m
747
748 {--------------------------------------------------------------------
749 @bin@ assures that we never have empty trees within a tree.
750 --------------------------------------------------------------------}
751 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
752 bin p m l Nil = l
753 bin p m Nil r = r
754 bin p m l r = Bin p m l r
755
756
757 {--------------------------------------------------------------------
758 Endian independent bit twiddling
759 --------------------------------------------------------------------}
760 zero :: Int -> Mask -> Bool
761 zero i m
762 = (natFromInt i) .&. (natFromInt m) == 0
763
764 nomatch,match :: Int -> Prefix -> Mask -> Bool
765 nomatch i p m
766 = (mask i m) /= p
767
768 match i p m
769 = (mask i m) == p
770
771 mask :: Int -> Mask -> Prefix
772 mask i m
773 = maskW (natFromInt i) (natFromInt m)
774
775 zeroN :: Nat -> Nat -> Bool
776 zeroN i m = (i .&. m) == 0
777
778 {--------------------------------------------------------------------
779 Big endian operations
780 --------------------------------------------------------------------}
781 maskW :: Nat -> Nat -> Prefix
782 maskW i m
783 = intFromNat (i .&. (complement (m-1) `xor` m))
784
785 shorter :: Mask -> Mask -> Bool
786 shorter m1 m2
787 = (natFromInt m1) > (natFromInt m2)
788
789 branchMask :: Prefix -> Prefix -> Mask
790 branchMask p1 p2
791 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
792
793 {----------------------------------------------------------------------
794 Finding the highest bit (mask) in a word [x] can be done efficiently in
795 three ways:
796 * convert to a floating point value and the mantissa tells us the
797 [log2(x)] that corresponds with the highest bit position. The mantissa
798 is retrieved either via the standard C function [frexp] or by some bit
799 twiddling on IEEE compatible numbers (float). Note that one needs to
800 use at least [double] precision for an accurate mantissa of 32 bit
801 numbers.
802 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
803 * use processor specific assembler instruction (asm).
804
805 The most portable way would be [bit], but is it efficient enough?
806 I have measured the cycle counts of the different methods on an AMD
807 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
808
809 highestBitMask: method cycles
810 --------------
811 frexp 200
812 float 33
813 bit 11
814 asm 12
815
816 highestBit: method cycles
817 --------------
818 frexp 195
819 float 33
820 bit 11
821 asm 11
822
823 Wow, the bit twiddling is on today's RISC like machines even faster
824 than a single CISC instruction (BSR)!
825 ----------------------------------------------------------------------}
826
827 {----------------------------------------------------------------------
828 [highestBitMask] returns a word where only the highest bit is set.
829 It is found by first setting all bits in lower positions than the
830 highest bit and than taking an exclusive or with the original value.
831 Allthough the function may look expensive, GHC compiles this into
832 excellent C code that subsequently compiled into highly efficient
833 machine code. The algorithm is derived from Jorg Arndt's FXT library.
834 ----------------------------------------------------------------------}
835 highestBitMask :: Nat -> Nat
836 highestBitMask x
837 = case (x .|. shiftRL x 1) of
838 x -> case (x .|. shiftRL x 2) of
839 x -> case (x .|. shiftRL x 4) of
840 x -> case (x .|. shiftRL x 8) of
841 x -> case (x .|. shiftRL x 16) of
842 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
843 x -> (x `xor` (shiftRL x 1))
844
845
846 {--------------------------------------------------------------------
847 Utilities
848 --------------------------------------------------------------------}
849 foldlStrict f z xs
850 = case xs of
851 [] -> z
852 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
853
854
855 {-
856 {--------------------------------------------------------------------
857 Testing
858 --------------------------------------------------------------------}
859 testTree :: [Int] -> IntSet
860 testTree xs = fromList xs
861 test1 = testTree [1..20]
862 test2 = testTree [30,29..10]
863 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
864
865 {--------------------------------------------------------------------
866 QuickCheck
867 --------------------------------------------------------------------}
868 qcheck prop
869 = check config prop
870 where
871 config = Config
872 { configMaxTest = 500
873 , configMaxFail = 5000
874 , configSize = \n -> (div n 2 + 3)
875 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
876 }
877
878
879 {--------------------------------------------------------------------
880 Arbitrary, reasonably balanced trees
881 --------------------------------------------------------------------}
882 instance Arbitrary IntSet where
883 arbitrary = do{ xs <- arbitrary
884 ; return (fromList xs)
885 }
886
887
888 {--------------------------------------------------------------------
889 Single, Insert, Delete
890 --------------------------------------------------------------------}
891 prop_Single :: Int -> Bool
892 prop_Single x
893 = (insert x empty == singleton x)
894
895 prop_InsertDelete :: Int -> IntSet -> Property
896 prop_InsertDelete k t
897 = not (member k t) ==> delete k (insert k t) == t
898
899
900 {--------------------------------------------------------------------
901 Union
902 --------------------------------------------------------------------}
903 prop_UnionInsert :: Int -> IntSet -> Bool
904 prop_UnionInsert x t
905 = union t (singleton x) == insert x t
906
907 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
908 prop_UnionAssoc t1 t2 t3
909 = union t1 (union t2 t3) == union (union t1 t2) t3
910
911 prop_UnionComm :: IntSet -> IntSet -> Bool
912 prop_UnionComm t1 t2
913 = (union t1 t2 == union t2 t1)
914
915 prop_Diff :: [Int] -> [Int] -> Bool
916 prop_Diff xs ys
917 = toAscList (difference (fromList xs) (fromList ys))
918 == List.sort ((List.\\) (nub xs) (nub ys))
919
920 prop_Int :: [Int] -> [Int] -> Bool
921 prop_Int xs ys
922 = toAscList (intersection (fromList xs) (fromList ys))
923 == List.sort (nub ((List.intersect) (xs) (ys)))
924
925 {--------------------------------------------------------------------
926 Lists
927 --------------------------------------------------------------------}
928 prop_Ordered
929 = forAll (choose (5,100)) $ \n ->
930 let xs = [0..n::Int]
931 in fromAscList xs == fromList xs
932
933 prop_List :: [Int] -> Bool
934 prop_List xs
935 = (sort (nub xs) == toAscList (fromList xs))
936 -}