expand advice on importing these modules
[packages/old-time.git] / Data / Map.hs
1 {-# OPTIONS_GHC -fno-bang-patterns #-}
2
3 -----------------------------------------------------------------------------
4 -- |
5 -- Module : Data.Map
6 -- Copyright : (c) Daan Leijen 2002
7 -- License : BSD-style
8 -- Maintainer : libraries@haskell.org
9 -- Stability : provisional
10 -- Portability : portable
11 --
12 -- An efficient implementation of maps from keys to values (dictionaries).
13 --
14 -- Since many function names (but not the type name) clash with
15 -- "Prelude" names, this module is usually imported @qualified@, e.g.
16 --
17 -- > import Data.Map (Map)
18 -- > import qualified Data.Map as Map
19 --
20 -- The implementation of 'Map' is based on /size balanced/ binary trees (or
21 -- trees of /bounded balance/) as described by:
22 --
23 -- * Stephen Adams, \"/Efficient sets: a balancing act/\",
24 -- Journal of Functional Programming 3(4):553-562, October 1993,
25 -- <http://www.swiss.ai.mit.edu/~adams/BB>.
26 --
27 -- * J. Nievergelt and E.M. Reingold,
28 -- \"/Binary search trees of bounded balance/\",
29 -- SIAM journal of computing 2(1), March 1973.
30 --
31 -- Note that the implementation is /left-biased/ -- the elements of a
32 -- first argument are always preferred to the second, for example in
33 -- 'union' or 'insert'.
34 -----------------------------------------------------------------------------
35
36 module Data.Map (
37 -- * Map type
38 Map -- instance Eq,Show,Read
39
40 -- * Operators
41 , (!), (\\)
42
43
44 -- * Query
45 , null
46 , size
47 , member
48 , notMember
49 , lookup
50 , findWithDefault
51
52 -- * Construction
53 , empty
54 , singleton
55
56 -- ** Insertion
57 , insert
58 , insertWith, insertWithKey, insertLookupWithKey
59
60 -- ** Delete\/Update
61 , delete
62 , adjust
63 , adjustWithKey
64 , update
65 , updateWithKey
66 , updateLookupWithKey
67 , alter
68
69 -- * Combine
70
71 -- ** Union
72 , union
73 , unionWith
74 , unionWithKey
75 , unions
76 , unionsWith
77
78 -- ** Difference
79 , difference
80 , differenceWith
81 , differenceWithKey
82
83 -- ** Intersection
84 , intersection
85 , intersectionWith
86 , intersectionWithKey
87
88 -- * Traversal
89 -- ** Map
90 , map
91 , mapWithKey
92 , mapAccum
93 , mapAccumWithKey
94 , mapKeys
95 , mapKeysWith
96 , mapKeysMonotonic
97
98 -- ** Fold
99 , fold
100 , foldWithKey
101
102 -- * Conversion
103 , elems
104 , keys
105 , keysSet
106 , assocs
107
108 -- ** Lists
109 , toList
110 , fromList
111 , fromListWith
112 , fromListWithKey
113
114 -- ** Ordered lists
115 , toAscList
116 , fromAscList
117 , fromAscListWith
118 , fromAscListWithKey
119 , fromDistinctAscList
120
121 -- * Filter
122 , filter
123 , filterWithKey
124 , partition
125 , partitionWithKey
126
127 , mapMaybe
128 , mapMaybeWithKey
129 , mapEither
130 , mapEitherWithKey
131
132 , split
133 , splitLookup
134
135 -- * Submap
136 , isSubmapOf, isSubmapOfBy
137 , isProperSubmapOf, isProperSubmapOfBy
138
139 -- * Indexed
140 , lookupIndex
141 , findIndex
142 , elemAt
143 , updateAt
144 , deleteAt
145
146 -- * Min\/Max
147 , findMin
148 , findMax
149 , deleteMin
150 , deleteMax
151 , deleteFindMin
152 , deleteFindMax
153 , updateMin
154 , updateMax
155 , updateMinWithKey
156 , updateMaxWithKey
157 , minView
158 , maxView
159
160 -- * Debugging
161 , showTree
162 , showTreeWith
163 , valid
164 ) where
165
166 import Prelude hiding (lookup,map,filter,foldr,foldl,null)
167 import qualified Data.Set as Set
168 import qualified Data.List as List
169 import Data.Monoid (Monoid(..))
170 import Data.Typeable
171 import Control.Applicative (Applicative(..), (<$>))
172 import Data.Traversable (Traversable(traverse))
173 import Data.Foldable (Foldable(foldMap))
174
175 {-
176 -- for quick check
177 import qualified Prelude
178 import qualified List
179 import Debug.QuickCheck
180 import List(nub,sort)
181 -}
182
183 #if __GLASGOW_HASKELL__
184 import Text.Read
185 import Data.Generics.Basics
186 import Data.Generics.Instances
187 #endif
188
189 {--------------------------------------------------------------------
190 Operators
191 --------------------------------------------------------------------}
192 infixl 9 !,\\ --
193
194 -- | /O(log n)/. Find the value at a key.
195 -- Calls 'error' when the element can not be found.
196 (!) :: Ord k => Map k a -> k -> a
197 m ! k = find k m
198
199 -- | /O(n+m)/. See 'difference'.
200 (\\) :: Ord k => Map k a -> Map k b -> Map k a
201 m1 \\ m2 = difference m1 m2
202
203 {--------------------------------------------------------------------
204 Size balanced trees.
205 --------------------------------------------------------------------}
206 -- | A Map from keys @k@ to values @a@.
207 data Map k a = Tip
208 | Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
209
210 type Size = Int
211
212 instance (Ord k) => Monoid (Map k v) where
213 mempty = empty
214 mappend = union
215 mconcat = unions
216
217 #if __GLASGOW_HASKELL__
218
219 {--------------------------------------------------------------------
220 A Data instance
221 --------------------------------------------------------------------}
222
223 -- This instance preserves data abstraction at the cost of inefficiency.
224 -- We omit reflection services for the sake of data abstraction.
225
226 instance (Data k, Data a, Ord k) => Data (Map k a) where
227 gfoldl f z map = z fromList `f` (toList map)
228 toConstr _ = error "toConstr"
229 gunfold _ _ = error "gunfold"
230 dataTypeOf _ = mkNorepType "Data.Map.Map"
231 dataCast2 f = gcast2 f
232
233 #endif
234
235 {--------------------------------------------------------------------
236 Query
237 --------------------------------------------------------------------}
238 -- | /O(1)/. Is the map empty?
239 null :: Map k a -> Bool
240 null t
241 = case t of
242 Tip -> True
243 Bin sz k x l r -> False
244
245 -- | /O(1)/. The number of elements in the map.
246 size :: Map k a -> Int
247 size t
248 = case t of
249 Tip -> 0
250 Bin sz k x l r -> sz
251
252
253 -- | /O(log n)/. Lookup the value at a key in the map.
254 lookup :: (Monad m,Ord k) => k -> Map k a -> m a
255 lookup k t = case lookup' k t of
256 Just x -> return x
257 Nothing -> fail "Data.Map.lookup: Key not found"
258 lookup' :: Ord k => k -> Map k a -> Maybe a
259 lookup' k t
260 = case t of
261 Tip -> Nothing
262 Bin sz kx x l r
263 -> case compare k kx of
264 LT -> lookup' k l
265 GT -> lookup' k r
266 EQ -> Just x
267
268 lookupAssoc :: Ord k => k -> Map k a -> Maybe (k,a)
269 lookupAssoc k t
270 = case t of
271 Tip -> Nothing
272 Bin sz kx x l r
273 -> case compare k kx of
274 LT -> lookupAssoc k l
275 GT -> lookupAssoc k r
276 EQ -> Just (kx,x)
277
278 -- | /O(log n)/. Is the key a member of the map?
279 member :: Ord k => k -> Map k a -> Bool
280 member k m
281 = case lookup k m of
282 Nothing -> False
283 Just x -> True
284
285 -- | /O(log n)/. Is the key not a member of the map?
286 notMember :: Ord k => k -> Map k a -> Bool
287 notMember k m = not $ member k m
288
289 -- | /O(log n)/. Find the value at a key.
290 -- Calls 'error' when the element can not be found.
291 find :: Ord k => k -> Map k a -> a
292 find k m
293 = case lookup k m of
294 Nothing -> error "Map.find: element not in the map"
295 Just x -> x
296
297 -- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns
298 -- the value at key @k@ or returns @def@ when the key is not in the map.
299 findWithDefault :: Ord k => a -> k -> Map k a -> a
300 findWithDefault def k m
301 = case lookup k m of
302 Nothing -> def
303 Just x -> x
304
305
306
307 {--------------------------------------------------------------------
308 Construction
309 --------------------------------------------------------------------}
310 -- | /O(1)/. The empty map.
311 empty :: Map k a
312 empty
313 = Tip
314
315 -- | /O(1)/. A map with a single element.
316 singleton :: k -> a -> Map k a
317 singleton k x
318 = Bin 1 k x Tip Tip
319
320 {--------------------------------------------------------------------
321 Insertion
322 --------------------------------------------------------------------}
323 -- | /O(log n)/. Insert a new key and value in the map.
324 -- If the key is already present in the map, the associated value is
325 -- replaced with the supplied value, i.e. 'insert' is equivalent to
326 -- @'insertWith' 'const'@.
327 insert :: Ord k => k -> a -> Map k a -> Map k a
328 insert kx x t
329 = case t of
330 Tip -> singleton kx x
331 Bin sz ky y l r
332 -> case compare kx ky of
333 LT -> balance ky y (insert kx x l) r
334 GT -> balance ky y l (insert kx x r)
335 EQ -> Bin sz kx x l r
336
337 -- | /O(log n)/. Insert with a combining function.
338 -- @'insertWith' f key value mp@
339 -- will insert the pair (key, value) into @mp@ if key does
340 -- not exist in the map. If the key does exist, the function will
341 -- insert the pair @(key, f new_value old_value)@.
342 insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
343 insertWith f k x m
344 = insertWithKey (\k x y -> f x y) k x m
345
346 -- | /O(log n)/. Insert with a combining function.
347 -- @'insertWithKey' f key value mp@
348 -- will insert the pair (key, value) into @mp@ if key does
349 -- not exist in the map. If the key does exist, the function will
350 -- insert the pair @(key,f key new_value old_value)@.
351 -- Note that the key passed to f is the same key passed to 'insertWithKey'.
352 insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
353 insertWithKey f kx x t
354 = case t of
355 Tip -> singleton kx x
356 Bin sy ky y l r
357 -> case compare kx ky of
358 LT -> balance ky y (insertWithKey f kx x l) r
359 GT -> balance ky y l (insertWithKey f kx x r)
360 EQ -> Bin sy kx (f kx x y) l r
361
362 -- | /O(log n)/. The expression (@'insertLookupWithKey' f k x map@)
363 -- is a pair where the first element is equal to (@'lookup' k map@)
364 -- and the second element equal to (@'insertWithKey' f k x map@).
365 insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a,Map k a)
366 insertLookupWithKey f kx x t
367 = case t of
368 Tip -> (Nothing, singleton kx x)
369 Bin sy ky y l r
370 -> case compare kx ky of
371 LT -> let (found,l') = insertLookupWithKey f kx x l in (found,balance ky y l' r)
372 GT -> let (found,r') = insertLookupWithKey f kx x r in (found,balance ky y l r')
373 EQ -> (Just y, Bin sy kx (f kx x y) l r)
374
375 {--------------------------------------------------------------------
376 Deletion
377 [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
378 --------------------------------------------------------------------}
379 -- | /O(log n)/. Delete a key and its value from the map. When the key is not
380 -- a member of the map, the original map is returned.
381 delete :: Ord k => k -> Map k a -> Map k a
382 delete k t
383 = case t of
384 Tip -> Tip
385 Bin sx kx x l r
386 -> case compare k kx of
387 LT -> balance kx x (delete k l) r
388 GT -> balance kx x l (delete k r)
389 EQ -> glue l r
390
391 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
392 -- a member of the map, the original map is returned.
393 adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
394 adjust f k m
395 = adjustWithKey (\k x -> f x) k m
396
397 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
398 -- a member of the map, the original map is returned.
399 adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
400 adjustWithKey f k m
401 = updateWithKey (\k x -> Just (f k x)) k m
402
403 -- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@
404 -- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
405 -- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
406 update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
407 update f k m
408 = updateWithKey (\k x -> f x) k m
409
410 -- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the
411 -- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',
412 -- the element is deleted. If it is (@'Just' y@), the key @k@ is bound
413 -- to the new value @y@.
414 updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
415 updateWithKey f k t
416 = case t of
417 Tip -> Tip
418 Bin sx kx x l r
419 -> case compare k kx of
420 LT -> balance kx x (updateWithKey f k l) r
421 GT -> balance kx x l (updateWithKey f k r)
422 EQ -> case f kx x of
423 Just x' -> Bin sx kx x' l r
424 Nothing -> glue l r
425
426 -- | /O(log n)/. Lookup and update.
427 updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
428 updateLookupWithKey f k t
429 = case t of
430 Tip -> (Nothing,Tip)
431 Bin sx kx x l r
432 -> case compare k kx of
433 LT -> let (found,l') = updateLookupWithKey f k l in (found,balance kx x l' r)
434 GT -> let (found,r') = updateLookupWithKey f k r in (found,balance kx x l r')
435 EQ -> case f kx x of
436 Just x' -> (Just x',Bin sx kx x' l r)
437 Nothing -> (Just x,glue l r)
438
439 -- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
440 -- 'alter' can be used to insert, delete, or update a value in a 'Map'.
441 -- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@
442 alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
443 alter f k t
444 = case t of
445 Tip -> case f Nothing of
446 Nothing -> Tip
447 Just x -> singleton k x
448 Bin sx kx x l r
449 -> case compare k kx of
450 LT -> balance kx x (alter f k l) r
451 GT -> balance kx x l (alter f k r)
452 EQ -> case f (Just x) of
453 Just x' -> Bin sx kx x' l r
454 Nothing -> glue l r
455
456 {--------------------------------------------------------------------
457 Indexing
458 --------------------------------------------------------------------}
459 -- | /O(log n)/. Return the /index/ of a key. The index is a number from
460 -- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
461 -- the key is not a 'member' of the map.
462 findIndex :: Ord k => k -> Map k a -> Int
463 findIndex k t
464 = case lookupIndex k t of
465 Nothing -> error "Map.findIndex: element is not in the map"
466 Just idx -> idx
467
468 -- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
469 -- /0/ up to, but not including, the 'size' of the map.
470 lookupIndex :: (Monad m,Ord k) => k -> Map k a -> m Int
471 lookupIndex k t = case lookup 0 t of
472 Nothing -> fail "Data.Map.lookupIndex: Key not found."
473 Just x -> return x
474 where
475 lookup idx Tip = Nothing
476 lookup idx (Bin _ kx x l r)
477 = case compare k kx of
478 LT -> lookup idx l
479 GT -> lookup (idx + size l + 1) r
480 EQ -> Just (idx + size l)
481
482 -- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
483 -- invalid index is used.
484 elemAt :: Int -> Map k a -> (k,a)
485 elemAt i Tip = error "Map.elemAt: index out of range"
486 elemAt i (Bin _ kx x l r)
487 = case compare i sizeL of
488 LT -> elemAt i l
489 GT -> elemAt (i-sizeL-1) r
490 EQ -> (kx,x)
491 where
492 sizeL = size l
493
494 -- | /O(log n)/. Update the element at /index/. Calls 'error' when an
495 -- invalid index is used.
496 updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
497 updateAt f i Tip = error "Map.updateAt: index out of range"
498 updateAt f i (Bin sx kx x l r)
499 = case compare i sizeL of
500 LT -> updateAt f i l
501 GT -> updateAt f (i-sizeL-1) r
502 EQ -> case f kx x of
503 Just x' -> Bin sx kx x' l r
504 Nothing -> glue l r
505 where
506 sizeL = size l
507
508 -- | /O(log n)/. Delete the element at /index/.
509 -- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).
510 deleteAt :: Int -> Map k a -> Map k a
511 deleteAt i map
512 = updateAt (\k x -> Nothing) i map
513
514
515 {--------------------------------------------------------------------
516 Minimal, Maximal
517 --------------------------------------------------------------------}
518 -- | /O(log n)/. The minimal key of the map.
519 findMin :: Map k a -> (k,a)
520 findMin (Bin _ kx x Tip r) = (kx,x)
521 findMin (Bin _ kx x l r) = findMin l
522 findMin Tip = error "Map.findMin: empty map has no minimal element"
523
524 -- | /O(log n)/. The maximal key of the map.
525 findMax :: Map k a -> (k,a)
526 findMax (Bin _ kx x l Tip) = (kx,x)
527 findMax (Bin _ kx x l r) = findMax r
528 findMax Tip = error "Map.findMax: empty map has no maximal element"
529
530 -- | /O(log n)/. Delete the minimal key.
531 deleteMin :: Map k a -> Map k a
532 deleteMin (Bin _ kx x Tip r) = r
533 deleteMin (Bin _ kx x l r) = balance kx x (deleteMin l) r
534 deleteMin Tip = Tip
535
536 -- | /O(log n)/. Delete the maximal key.
537 deleteMax :: Map k a -> Map k a
538 deleteMax (Bin _ kx x l Tip) = l
539 deleteMax (Bin _ kx x l r) = balance kx x l (deleteMax r)
540 deleteMax Tip = Tip
541
542 -- | /O(log n)/. Update the value at the minimal key.
543 updateMin :: (a -> Maybe a) -> Map k a -> Map k a
544 updateMin f m
545 = updateMinWithKey (\k x -> f x) m
546
547 -- | /O(log n)/. Update the value at the maximal key.
548 updateMax :: (a -> Maybe a) -> Map k a -> Map k a
549 updateMax f m
550 = updateMaxWithKey (\k x -> f x) m
551
552
553 -- | /O(log n)/. Update the value at the minimal key.
554 updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
555 updateMinWithKey f t
556 = case t of
557 Bin sx kx x Tip r -> case f kx x of
558 Nothing -> r
559 Just x' -> Bin sx kx x' Tip r
560 Bin sx kx x l r -> balance kx x (updateMinWithKey f l) r
561 Tip -> Tip
562
563 -- | /O(log n)/. Update the value at the maximal key.
564 updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
565 updateMaxWithKey f t
566 = case t of
567 Bin sx kx x l Tip -> case f kx x of
568 Nothing -> l
569 Just x' -> Bin sx kx x' l Tip
570 Bin sx kx x l r -> balance kx x l (updateMaxWithKey f r)
571 Tip -> Tip
572
573 -- | /O(log n)/. Retrieves the minimal key of the map, and the map stripped from that element
574 -- @fail@s (in the monad) when passed an empty map.
575 minView :: Monad m => Map k a -> m (Map k a, (k,a))
576 minView Tip = fail "Map.minView: empty map"
577 minView x = return (swap $ deleteFindMin x)
578
579 -- | /O(log n)/. Retrieves the maximal key of the map, and the map stripped from that element
580 -- @fail@s (in the monad) when passed an empty map.
581 maxView :: Monad m => Map k a -> m (Map k a, (k,a))
582 maxView Tip = fail "Map.maxView: empty map"
583 maxView x = return (swap $ deleteFindMax x)
584
585 swap (a,b) = (b,a)
586
587 {--------------------------------------------------------------------
588 Union.
589 --------------------------------------------------------------------}
590 -- | The union of a list of maps:
591 -- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).
592 unions :: Ord k => [Map k a] -> Map k a
593 unions ts
594 = foldlStrict union empty ts
595
596 -- | The union of a list of maps, with a combining operation:
597 -- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).
598 unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a
599 unionsWith f ts
600 = foldlStrict (unionWith f) empty ts
601
602 -- | /O(n+m)/.
603 -- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.
604 -- It prefers @t1@ when duplicate keys are encountered,
605 -- i.e. (@'union' == 'unionWith' 'const'@).
606 -- The implementation uses the efficient /hedge-union/ algorithm.
607 -- Hedge-union is more efficient on (bigset `union` smallset)
608 union :: Ord k => Map k a -> Map k a -> Map k a
609 union Tip t2 = t2
610 union t1 Tip = t1
611 union t1 t2 = hedgeUnionL (const LT) (const GT) t1 t2
612
613 -- left-biased hedge union
614 hedgeUnionL cmplo cmphi t1 Tip
615 = t1
616 hedgeUnionL cmplo cmphi Tip (Bin _ kx x l r)
617 = join kx x (filterGt cmplo l) (filterLt cmphi r)
618 hedgeUnionL cmplo cmphi (Bin _ kx x l r) t2
619 = join kx x (hedgeUnionL cmplo cmpkx l (trim cmplo cmpkx t2))
620 (hedgeUnionL cmpkx cmphi r (trim cmpkx cmphi t2))
621 where
622 cmpkx k = compare kx k
623
624 -- right-biased hedge union
625 hedgeUnionR cmplo cmphi t1 Tip
626 = t1
627 hedgeUnionR cmplo cmphi Tip (Bin _ kx x l r)
628 = join kx x (filterGt cmplo l) (filterLt cmphi r)
629 hedgeUnionR cmplo cmphi (Bin _ kx x l r) t2
630 = join kx newx (hedgeUnionR cmplo cmpkx l lt)
631 (hedgeUnionR cmpkx cmphi r gt)
632 where
633 cmpkx k = compare kx k
634 lt = trim cmplo cmpkx t2
635 (found,gt) = trimLookupLo kx cmphi t2
636 newx = case found of
637 Nothing -> x
638 Just (_,y) -> y
639
640 {--------------------------------------------------------------------
641 Union with a combining function
642 --------------------------------------------------------------------}
643 -- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
644 unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
645 unionWith f m1 m2
646 = unionWithKey (\k x y -> f x y) m1 m2
647
648 -- | /O(n+m)/.
649 -- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
650 -- Hedge-union is more efficient on (bigset `union` smallset).
651 unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
652 unionWithKey f Tip t2 = t2
653 unionWithKey f t1 Tip = t1
654 unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
655
656 hedgeUnionWithKey f cmplo cmphi t1 Tip
657 = t1
658 hedgeUnionWithKey f cmplo cmphi Tip (Bin _ kx x l r)
659 = join kx x (filterGt cmplo l) (filterLt cmphi r)
660 hedgeUnionWithKey f cmplo cmphi (Bin _ kx x l r) t2
661 = join kx newx (hedgeUnionWithKey f cmplo cmpkx l lt)
662 (hedgeUnionWithKey f cmpkx cmphi r gt)
663 where
664 cmpkx k = compare kx k
665 lt = trim cmplo cmpkx t2
666 (found,gt) = trimLookupLo kx cmphi t2
667 newx = case found of
668 Nothing -> x
669 Just (_,y) -> f kx x y
670
671 {--------------------------------------------------------------------
672 Difference
673 --------------------------------------------------------------------}
674 -- | /O(n+m)/. Difference of two maps.
675 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
676 difference :: Ord k => Map k a -> Map k b -> Map k a
677 difference Tip t2 = Tip
678 difference t1 Tip = t1
679 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
680
681 hedgeDiff cmplo cmphi Tip t
682 = Tip
683 hedgeDiff cmplo cmphi (Bin _ kx x l r) Tip
684 = join kx x (filterGt cmplo l) (filterLt cmphi r)
685 hedgeDiff cmplo cmphi t (Bin _ kx x l r)
686 = merge (hedgeDiff cmplo cmpkx (trim cmplo cmpkx t) l)
687 (hedgeDiff cmpkx cmphi (trim cmpkx cmphi t) r)
688 where
689 cmpkx k = compare kx k
690
691 -- | /O(n+m)/. Difference with a combining function.
692 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
693 differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
694 differenceWith f m1 m2
695 = differenceWithKey (\k x y -> f x y) m1 m2
696
697 -- | /O(n+m)/. Difference with a combining function. When two equal keys are
698 -- encountered, the combining function is applied to the key and both values.
699 -- If it returns 'Nothing', the element is discarded (proper set difference). If
700 -- it returns (@'Just' y@), the element is updated with a new value @y@.
701 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
702 differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
703 differenceWithKey f Tip t2 = Tip
704 differenceWithKey f t1 Tip = t1
705 differenceWithKey f t1 t2 = hedgeDiffWithKey f (const LT) (const GT) t1 t2
706
707 hedgeDiffWithKey f cmplo cmphi Tip t
708 = Tip
709 hedgeDiffWithKey f cmplo cmphi (Bin _ kx x l r) Tip
710 = join kx x (filterGt cmplo l) (filterLt cmphi r)
711 hedgeDiffWithKey f cmplo cmphi t (Bin _ kx x l r)
712 = case found of
713 Nothing -> merge tl tr
714 Just (ky,y) ->
715 case f ky y x of
716 Nothing -> merge tl tr
717 Just z -> join ky z tl tr
718 where
719 cmpkx k = compare kx k
720 lt = trim cmplo cmpkx t
721 (found,gt) = trimLookupLo kx cmphi t
722 tl = hedgeDiffWithKey f cmplo cmpkx lt l
723 tr = hedgeDiffWithKey f cmpkx cmphi gt r
724
725
726
727 {--------------------------------------------------------------------
728 Intersection
729 --------------------------------------------------------------------}
730 -- | /O(n+m)/. Intersection of two maps. The values in the first
731 -- map are returned, i.e. (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).
732 intersection :: Ord k => Map k a -> Map k b -> Map k a
733 intersection m1 m2
734 = intersectionWithKey (\k x y -> x) m1 m2
735
736 -- | /O(n+m)/. Intersection with a combining function.
737 intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
738 intersectionWith f m1 m2
739 = intersectionWithKey (\k x y -> f x y) m1 m2
740
741 -- | /O(n+m)/. Intersection with a combining function.
742 -- Intersection is more efficient on (bigset `intersection` smallset)
743 --intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
744 --intersectionWithKey f Tip t = Tip
745 --intersectionWithKey f t Tip = Tip
746 --intersectionWithKey f t1 t2 = intersectWithKey f t1 t2
747 --
748 --intersectWithKey f Tip t = Tip
749 --intersectWithKey f t Tip = Tip
750 --intersectWithKey f t (Bin _ kx x l r)
751 -- = case found of
752 -- Nothing -> merge tl tr
753 -- Just y -> join kx (f kx y x) tl tr
754 -- where
755 -- (lt,found,gt) = splitLookup kx t
756 -- tl = intersectWithKey f lt l
757 -- tr = intersectWithKey f gt r
758
759
760 intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
761 intersectionWithKey f Tip t = Tip
762 intersectionWithKey f t Tip = Tip
763 intersectionWithKey f t1@(Bin s1 k1 x1 l1 r1) t2@(Bin s2 k2 x2 l2 r2) =
764 if s1 >= s2 then
765 let (lt,found,gt) = splitLookupWithKey k2 t1
766 tl = intersectionWithKey f lt l2
767 tr = intersectionWithKey f gt r2
768 in case found of
769 Just (k,x) -> join k (f k x x2) tl tr
770 Nothing -> merge tl tr
771 else let (lt,found,gt) = splitLookup k1 t2
772 tl = intersectionWithKey f l1 lt
773 tr = intersectionWithKey f r1 gt
774 in case found of
775 Just x -> join k1 (f k1 x1 x) tl tr
776 Nothing -> merge tl tr
777
778
779
780 {--------------------------------------------------------------------
781 Submap
782 --------------------------------------------------------------------}
783 -- | /O(n+m)/.
784 -- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
785 isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
786 isSubmapOf m1 m2
787 = isSubmapOfBy (==) m1 m2
788
789 {- | /O(n+m)/.
790 The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if
791 all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when
792 applied to their respective values. For example, the following
793 expressions are all 'True':
794
795 > isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
796 > isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
797 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
798
799 But the following are all 'False':
800
801 > isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
802 > isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
803 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
804 -}
805 isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool
806 isSubmapOfBy f t1 t2
807 = (size t1 <= size t2) && (submap' f t1 t2)
808
809 submap' f Tip t = True
810 submap' f t Tip = False
811 submap' f (Bin _ kx x l r) t
812 = case found of
813 Nothing -> False
814 Just y -> f x y && submap' f l lt && submap' f r gt
815 where
816 (lt,found,gt) = splitLookup kx t
817
818 -- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
819 -- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
820 isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
821 isProperSubmapOf m1 m2
822 = isProperSubmapOfBy (==) m1 m2
823
824 {- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
825 The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
826 @m1@ and @m2@ are not equal,
827 all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
828 applied to their respective values. For example, the following
829 expressions are all 'True':
830
831 > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
832 > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
833
834 But the following are all 'False':
835
836 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
837 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
838 > isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
839 -}
840 isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
841 isProperSubmapOfBy f t1 t2
842 = (size t1 < size t2) && (submap' f t1 t2)
843
844 {--------------------------------------------------------------------
845 Filter and partition
846 --------------------------------------------------------------------}
847 -- | /O(n)/. Filter all values that satisfy the predicate.
848 filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
849 filter p m
850 = filterWithKey (\k x -> p x) m
851
852 -- | /O(n)/. Filter all keys\/values that satisfy the predicate.
853 filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
854 filterWithKey p Tip = Tip
855 filterWithKey p (Bin _ kx x l r)
856 | p kx x = join kx x (filterWithKey p l) (filterWithKey p r)
857 | otherwise = merge (filterWithKey p l) (filterWithKey p r)
858
859
860 -- | /O(n)/. partition the map according to a predicate. The first
861 -- map contains all elements that satisfy the predicate, the second all
862 -- elements that fail the predicate. See also 'split'.
863 partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a)
864 partition p m
865 = partitionWithKey (\k x -> p x) m
866
867 -- | /O(n)/. partition the map according to a predicate. The first
868 -- map contains all elements that satisfy the predicate, the second all
869 -- elements that fail the predicate. See also 'split'.
870 partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
871 partitionWithKey p Tip = (Tip,Tip)
872 partitionWithKey p (Bin _ kx x l r)
873 | p kx x = (join kx x l1 r1,merge l2 r2)
874 | otherwise = (merge l1 r1,join kx x l2 r2)
875 where
876 (l1,l2) = partitionWithKey p l
877 (r1,r2) = partitionWithKey p r
878
879 -- | /O(n)/. Map values and collect the 'Just' results.
880 mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b
881 mapMaybe f m
882 = mapMaybeWithKey (\k x -> f x) m
883
884 -- | /O(n)/. Map keys\/values and collect the 'Just' results.
885 mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b
886 mapMaybeWithKey f Tip = Tip
887 mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of
888 Just y -> join kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
889 Nothing -> merge (mapMaybeWithKey f l) (mapMaybeWithKey f r)
890
891 -- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
892 mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)
893 mapEither f m
894 = mapEitherWithKey (\k x -> f x) m
895
896 -- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
897 mapEitherWithKey :: Ord k =>
898 (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
899 mapEitherWithKey f Tip = (Tip, Tip)
900 mapEitherWithKey f (Bin _ kx x l r) = case f kx x of
901 Left y -> (join kx y l1 r1, merge l2 r2)
902 Right z -> (merge l1 r1, join kx z l2 r2)
903 where
904 (l1,l2) = mapEitherWithKey f l
905 (r1,r2) = mapEitherWithKey f r
906
907 {--------------------------------------------------------------------
908 Mapping
909 --------------------------------------------------------------------}
910 -- | /O(n)/. Map a function over all values in the map.
911 map :: (a -> b) -> Map k a -> Map k b
912 map f m
913 = mapWithKey (\k x -> f x) m
914
915 -- | /O(n)/. Map a function over all values in the map.
916 mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
917 mapWithKey f Tip = Tip
918 mapWithKey f (Bin sx kx x l r)
919 = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)
920
921 -- | /O(n)/. The function 'mapAccum' threads an accumulating
922 -- argument through the map in ascending order of keys.
923 mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
924 mapAccum f a m
925 = mapAccumWithKey (\a k x -> f a x) a m
926
927 -- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating
928 -- argument through the map in ascending order of keys.
929 mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
930 mapAccumWithKey f a t
931 = mapAccumL f a t
932
933 -- | /O(n)/. The function 'mapAccumL' threads an accumulating
934 -- argument throught the map in ascending order of keys.
935 mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
936 mapAccumL f a t
937 = case t of
938 Tip -> (a,Tip)
939 Bin sx kx x l r
940 -> let (a1,l') = mapAccumL f a l
941 (a2,x') = f a1 kx x
942 (a3,r') = mapAccumL f a2 r
943 in (a3,Bin sx kx x' l' r')
944
945 -- | /O(n)/. The function 'mapAccumR' threads an accumulating
946 -- argument throught the map in descending order of keys.
947 mapAccumR :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
948 mapAccumR f a t
949 = case t of
950 Tip -> (a,Tip)
951 Bin sx kx x l r
952 -> let (a1,r') = mapAccumR f a r
953 (a2,x') = f a1 kx x
954 (a3,l') = mapAccumR f a2 l
955 in (a3,Bin sx kx x' l' r')
956
957 -- | /O(n*log n)/.
958 -- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
959 --
960 -- The size of the result may be smaller if @f@ maps two or more distinct
961 -- keys to the same new key. In this case the value at the smallest of
962 -- these keys is retained.
963
964 mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
965 mapKeys = mapKeysWith (\x y->x)
966
967 -- | /O(n*log n)/.
968 -- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
969 --
970 -- The size of the result may be smaller if @f@ maps two or more distinct
971 -- keys to the same new key. In this case the associated values will be
972 -- combined using @c@.
973
974 mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a
975 mapKeysWith c f = fromListWith c . List.map fFirst . toList
976 where fFirst (x,y) = (f x, y)
977
978
979 -- | /O(n)/.
980 -- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
981 -- is strictly monotonic.
982 -- /The precondition is not checked./
983 -- Semi-formally, we have:
984 --
985 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
986 -- > ==> mapKeysMonotonic f s == mapKeys f s
987 -- > where ls = keys s
988
989 mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a
990 mapKeysMonotonic f Tip = Tip
991 mapKeysMonotonic f (Bin sz k x l r) =
992 Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)
993
994 {--------------------------------------------------------------------
995 Folds
996 --------------------------------------------------------------------}
997
998 -- | /O(n)/. Fold the values in the map, such that
999 -- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.
1000 -- For example,
1001 --
1002 -- > elems map = fold (:) [] map
1003 --
1004 fold :: (a -> b -> b) -> b -> Map k a -> b
1005 fold f z m
1006 = foldWithKey (\k x z -> f x z) z m
1007
1008 -- | /O(n)/. Fold the keys and values in the map, such that
1009 -- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
1010 -- For example,
1011 --
1012 -- > keys map = foldWithKey (\k x ks -> k:ks) [] map
1013 --
1014 foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
1015 foldWithKey f z t
1016 = foldr f z t
1017
1018 -- | /O(n)/. In-order fold.
1019 foldi :: (k -> a -> b -> b -> b) -> b -> Map k a -> b
1020 foldi f z Tip = z
1021 foldi f z (Bin _ kx x l r) = f kx x (foldi f z l) (foldi f z r)
1022
1023 -- | /O(n)/. Post-order fold.
1024 foldr :: (k -> a -> b -> b) -> b -> Map k a -> b
1025 foldr f z Tip = z
1026 foldr f z (Bin _ kx x l r) = foldr f (f kx x (foldr f z r)) l
1027
1028 -- | /O(n)/. Pre-order fold.
1029 foldl :: (b -> k -> a -> b) -> b -> Map k a -> b
1030 foldl f z Tip = z
1031 foldl f z (Bin _ kx x l r) = foldl f (f (foldl f z l) kx x) r
1032
1033 {--------------------------------------------------------------------
1034 List variations
1035 --------------------------------------------------------------------}
1036 -- | /O(n)/.
1037 -- Return all elements of the map in the ascending order of their keys.
1038 elems :: Map k a -> [a]
1039 elems m
1040 = [x | (k,x) <- assocs m]
1041
1042 -- | /O(n)/. Return all keys of the map in ascending order.
1043 keys :: Map k a -> [k]
1044 keys m
1045 = [k | (k,x) <- assocs m]
1046
1047 -- | /O(n)/. The set of all keys of the map.
1048 keysSet :: Map k a -> Set.Set k
1049 keysSet m = Set.fromDistinctAscList (keys m)
1050
1051 -- | /O(n)/. Return all key\/value pairs in the map in ascending key order.
1052 assocs :: Map k a -> [(k,a)]
1053 assocs m
1054 = toList m
1055
1056 {--------------------------------------------------------------------
1057 Lists
1058 use [foldlStrict] to reduce demand on the control-stack
1059 --------------------------------------------------------------------}
1060 -- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
1061 fromList :: Ord k => [(k,a)] -> Map k a
1062 fromList xs
1063 = foldlStrict ins empty xs
1064 where
1065 ins t (k,x) = insert k x t
1066
1067 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
1068 fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a
1069 fromListWith f xs
1070 = fromListWithKey (\k x y -> f x y) xs
1071
1072 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
1073 fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1074 fromListWithKey f xs
1075 = foldlStrict ins empty xs
1076 where
1077 ins t (k,x) = insertWithKey f k x t
1078
1079 -- | /O(n)/. Convert to a list of key\/value pairs.
1080 toList :: Map k a -> [(k,a)]
1081 toList t = toAscList t
1082
1083 -- | /O(n)/. Convert to an ascending list.
1084 toAscList :: Map k a -> [(k,a)]
1085 toAscList t = foldr (\k x xs -> (k,x):xs) [] t
1086
1087 -- | /O(n)/.
1088 toDescList :: Map k a -> [(k,a)]
1089 toDescList t = foldl (\xs k x -> (k,x):xs) [] t
1090
1091
1092 {--------------------------------------------------------------------
1093 Building trees from ascending/descending lists can be done in linear time.
1094
1095 Note that if [xs] is ascending that:
1096 fromAscList xs == fromList xs
1097 fromAscListWith f xs == fromListWith f xs
1098 --------------------------------------------------------------------}
1099 -- | /O(n)/. Build a map from an ascending list in linear time.
1100 -- /The precondition (input list is ascending) is not checked./
1101 fromAscList :: Eq k => [(k,a)] -> Map k a
1102 fromAscList xs
1103 = fromAscListWithKey (\k x y -> x) xs
1104
1105 -- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
1106 -- /The precondition (input list is ascending) is not checked./
1107 fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
1108 fromAscListWith f xs
1109 = fromAscListWithKey (\k x y -> f x y) xs
1110
1111 -- | /O(n)/. Build a map from an ascending list in linear time with a
1112 -- combining function for equal keys.
1113 -- /The precondition (input list is ascending) is not checked./
1114 fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1115 fromAscListWithKey f xs
1116 = fromDistinctAscList (combineEq f xs)
1117 where
1118 -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
1119 combineEq f xs
1120 = case xs of
1121 [] -> []
1122 [x] -> [x]
1123 (x:xx) -> combineEq' x xx
1124
1125 combineEq' z [] = [z]
1126 combineEq' z@(kz,zz) (x@(kx,xx):xs)
1127 | kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs
1128 | otherwise = z:combineEq' x xs
1129
1130
1131 -- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
1132 -- /The precondition is not checked./
1133 fromDistinctAscList :: [(k,a)] -> Map k a
1134 fromDistinctAscList xs
1135 = build const (length xs) xs
1136 where
1137 -- 1) use continutations so that we use heap space instead of stack space.
1138 -- 2) special case for n==5 to build bushier trees.
1139 build c 0 xs = c Tip xs
1140 build c 5 xs = case xs of
1141 ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
1142 -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
1143 build c n xs = seq nr $ build (buildR nr c) nl xs
1144 where
1145 nl = n `div` 2
1146 nr = n - nl - 1
1147
1148 buildR n c l ((k,x):ys) = build (buildB l k x c) n ys
1149 buildB l k x c r zs = c (bin k x l r) zs
1150
1151
1152
1153 {--------------------------------------------------------------------
1154 Utility functions that return sub-ranges of the original
1155 tree. Some functions take a comparison function as argument to
1156 allow comparisons against infinite values. A function [cmplo k]
1157 should be read as [compare lo k].
1158
1159 [trim cmplo cmphi t] A tree that is either empty or where [cmplo k == LT]
1160 and [cmphi k == GT] for the key [k] of the root.
1161 [filterGt cmp t] A tree where for all keys [k]. [cmp k == LT]
1162 [filterLt cmp t] A tree where for all keys [k]. [cmp k == GT]
1163
1164 [split k t] Returns two trees [l] and [r] where all keys
1165 in [l] are <[k] and all keys in [r] are >[k].
1166 [splitLookup k t] Just like [split] but also returns whether [k]
1167 was found in the tree.
1168 --------------------------------------------------------------------}
1169
1170 {--------------------------------------------------------------------
1171 [trim lo hi t] trims away all subtrees that surely contain no
1172 values between the range [lo] to [hi]. The returned tree is either
1173 empty or the key of the root is between @lo@ and @hi@.
1174 --------------------------------------------------------------------}
1175 trim :: (k -> Ordering) -> (k -> Ordering) -> Map k a -> Map k a
1176 trim cmplo cmphi Tip = Tip
1177 trim cmplo cmphi t@(Bin sx kx x l r)
1178 = case cmplo kx of
1179 LT -> case cmphi kx of
1180 GT -> t
1181 le -> trim cmplo cmphi l
1182 ge -> trim cmplo cmphi r
1183
1184 trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a)
1185 trimLookupLo lo cmphi Tip = (Nothing,Tip)
1186 trimLookupLo lo cmphi t@(Bin sx kx x l r)
1187 = case compare lo kx of
1188 LT -> case cmphi kx of
1189 GT -> (lookupAssoc lo t, t)
1190 le -> trimLookupLo lo cmphi l
1191 GT -> trimLookupLo lo cmphi r
1192 EQ -> (Just (kx,x),trim (compare lo) cmphi r)
1193
1194
1195 {--------------------------------------------------------------------
1196 [filterGt k t] filter all keys >[k] from tree [t]
1197 [filterLt k t] filter all keys <[k] from tree [t]
1198 --------------------------------------------------------------------}
1199 filterGt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1200 filterGt cmp Tip = Tip
1201 filterGt cmp (Bin sx kx x l r)
1202 = case cmp kx of
1203 LT -> join kx x (filterGt cmp l) r
1204 GT -> filterGt cmp r
1205 EQ -> r
1206
1207 filterLt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1208 filterLt cmp Tip = Tip
1209 filterLt cmp (Bin sx kx x l r)
1210 = case cmp kx of
1211 LT -> filterLt cmp l
1212 GT -> join kx x l (filterLt cmp r)
1213 EQ -> l
1214
1215 {--------------------------------------------------------------------
1216 Split
1217 --------------------------------------------------------------------}
1218 -- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where
1219 -- the keys in @map1@ are smaller than @k@ and the keys in @map2@ larger than @k@. Any key equal to @k@ is found in neither @map1@ nor @map2@.
1220 split :: Ord k => k -> Map k a -> (Map k a,Map k a)
1221 split k Tip = (Tip,Tip)
1222 split k (Bin sx kx x l r)
1223 = case compare k kx of
1224 LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
1225 GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
1226 EQ -> (l,r)
1227
1228 -- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just
1229 -- like 'split' but also returns @'lookup' k map@.
1230 splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)
1231 splitLookup k Tip = (Tip,Nothing,Tip)
1232 splitLookup k (Bin sx kx x l r)
1233 = case compare k kx of
1234 LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
1235 GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
1236 EQ -> (l,Just x,r)
1237
1238 -- | /O(log n)/.
1239 splitLookupWithKey :: Ord k => k -> Map k a -> (Map k a,Maybe (k,a),Map k a)
1240 splitLookupWithKey k Tip = (Tip,Nothing,Tip)
1241 splitLookupWithKey k (Bin sx kx x l r)
1242 = case compare k kx of
1243 LT -> let (lt,z,gt) = splitLookupWithKey k l in (lt,z,join kx x gt r)
1244 GT -> let (lt,z,gt) = splitLookupWithKey k r in (join kx x l lt,z,gt)
1245 EQ -> (l,Just (kx, x),r)
1246
1247 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
1248 -- element was found in the original set.
1249 splitMember :: Ord k => k -> Map k a -> (Map k a,Bool,Map k a)
1250 splitMember x t = let (l,m,r) = splitLookup x t in
1251 (l,maybe False (const True) m,r)
1252
1253
1254 {--------------------------------------------------------------------
1255 Utility functions that maintain the balance properties of the tree.
1256 All constructors assume that all values in [l] < [k] and all values
1257 in [r] > [k], and that [l] and [r] are valid trees.
1258
1259 In order of sophistication:
1260 [Bin sz k x l r] The type constructor.
1261 [bin k x l r] Maintains the correct size, assumes that both [l]
1262 and [r] are balanced with respect to each other.
1263 [balance k x l r] Restores the balance and size.
1264 Assumes that the original tree was balanced and
1265 that [l] or [r] has changed by at most one element.
1266 [join k x l r] Restores balance and size.
1267
1268 Furthermore, we can construct a new tree from two trees. Both operations
1269 assume that all values in [l] < all values in [r] and that [l] and [r]
1270 are valid:
1271 [glue l r] Glues [l] and [r] together. Assumes that [l] and
1272 [r] are already balanced with respect to each other.
1273 [merge l r] Merges two trees and restores balance.
1274
1275 Note: in contrast to Adam's paper, we use (<=) comparisons instead
1276 of (<) comparisons in [join], [merge] and [balance].
1277 Quickcheck (on [difference]) showed that this was necessary in order
1278 to maintain the invariants. It is quite unsatisfactory that I haven't
1279 been able to find out why this is actually the case! Fortunately, it
1280 doesn't hurt to be a bit more conservative.
1281 --------------------------------------------------------------------}
1282
1283 {--------------------------------------------------------------------
1284 Join
1285 --------------------------------------------------------------------}
1286 join :: Ord k => k -> a -> Map k a -> Map k a -> Map k a
1287 join kx x Tip r = insertMin kx x r
1288 join kx x l Tip = insertMax kx x l
1289 join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
1290 | delta*sizeL <= sizeR = balance kz z (join kx x l lz) rz
1291 | delta*sizeR <= sizeL = balance ky y ly (join kx x ry r)
1292 | otherwise = bin kx x l r
1293
1294
1295 -- insertMin and insertMax don't perform potentially expensive comparisons.
1296 insertMax,insertMin :: k -> a -> Map k a -> Map k a
1297 insertMax kx x t
1298 = case t of
1299 Tip -> singleton kx x
1300 Bin sz ky y l r
1301 -> balance ky y l (insertMax kx x r)
1302
1303 insertMin kx x t
1304 = case t of
1305 Tip -> singleton kx x
1306 Bin sz ky y l r
1307 -> balance ky y (insertMin kx x l) r
1308
1309 {--------------------------------------------------------------------
1310 [merge l r]: merges two trees.
1311 --------------------------------------------------------------------}
1312 merge :: Map k a -> Map k a -> Map k a
1313 merge Tip r = r
1314 merge l Tip = l
1315 merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
1316 | delta*sizeL <= sizeR = balance ky y (merge l ly) ry
1317 | delta*sizeR <= sizeL = balance kx x lx (merge rx r)
1318 | otherwise = glue l r
1319
1320 {--------------------------------------------------------------------
1321 [glue l r]: glues two trees together.
1322 Assumes that [l] and [r] are already balanced with respect to each other.
1323 --------------------------------------------------------------------}
1324 glue :: Map k a -> Map k a -> Map k a
1325 glue Tip r = r
1326 glue l Tip = l
1327 glue l r
1328 | size l > size r = let ((km,m),l') = deleteFindMax l in balance km m l' r
1329 | otherwise = let ((km,m),r') = deleteFindMin r in balance km m l r'
1330
1331
1332 -- | /O(log n)/. Delete and find the minimal element.
1333 deleteFindMin :: Map k a -> ((k,a),Map k a)
1334 deleteFindMin t
1335 = case t of
1336 Bin _ k x Tip r -> ((k,x),r)
1337 Bin _ k x l r -> let (km,l') = deleteFindMin l in (km,balance k x l' r)
1338 Tip -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)
1339
1340 -- | /O(log n)/. Delete and find the maximal element.
1341 deleteFindMax :: Map k a -> ((k,a),Map k a)
1342 deleteFindMax t
1343 = case t of
1344 Bin _ k x l Tip -> ((k,x),l)
1345 Bin _ k x l r -> let (km,r') = deleteFindMax r in (km,balance k x l r')
1346 Tip -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
1347
1348
1349 {--------------------------------------------------------------------
1350 [balance l x r] balances two trees with value x.
1351 The sizes of the trees should balance after decreasing the
1352 size of one of them. (a rotation).
1353
1354 [delta] is the maximal relative difference between the sizes of
1355 two trees, it corresponds with the [w] in Adams' paper.
1356 [ratio] is the ratio between an outer and inner sibling of the
1357 heavier subtree in an unbalanced setting. It determines
1358 whether a double or single rotation should be performed
1359 to restore balance. It is correspondes with the inverse
1360 of $\alpha$ in Adam's article.
1361
1362 Note that:
1363 - [delta] should be larger than 4.646 with a [ratio] of 2.
1364 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
1365
1366 - A lower [delta] leads to a more 'perfectly' balanced tree.
1367 - A higher [delta] performs less rebalancing.
1368
1369 - Balancing is automatic for random data and a balancing
1370 scheme is only necessary to avoid pathological worst cases.
1371 Almost any choice will do, and in practice, a rather large
1372 [delta] may perform better than smaller one.
1373
1374 Note: in contrast to Adam's paper, we use a ratio of (at least) [2]
1375 to decide whether a single or double rotation is needed. Allthough
1376 he actually proves that this ratio is needed to maintain the
1377 invariants, his implementation uses an invalid ratio of [1].
1378 --------------------------------------------------------------------}
1379 delta,ratio :: Int
1380 delta = 5
1381 ratio = 2
1382
1383 balance :: k -> a -> Map k a -> Map k a -> Map k a
1384 balance k x l r
1385 | sizeL + sizeR <= 1 = Bin sizeX k x l r
1386 | sizeR >= delta*sizeL = rotateL k x l r
1387 | sizeL >= delta*sizeR = rotateR k x l r
1388 | otherwise = Bin sizeX k x l r
1389 where
1390 sizeL = size l
1391 sizeR = size r
1392 sizeX = sizeL + sizeR + 1
1393
1394 -- rotate
1395 rotateL k x l r@(Bin _ _ _ ly ry)
1396 | size ly < ratio*size ry = singleL k x l r
1397 | otherwise = doubleL k x l r
1398
1399 rotateR k x l@(Bin _ _ _ ly ry) r
1400 | size ry < ratio*size ly = singleR k x l r
1401 | otherwise = doubleR k x l r
1402
1403 -- basic rotations
1404 singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
1405 singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)
1406
1407 doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4) = bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)
1408 doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)
1409
1410
1411 {--------------------------------------------------------------------
1412 The bin constructor maintains the size of the tree
1413 --------------------------------------------------------------------}
1414 bin :: k -> a -> Map k a -> Map k a -> Map k a
1415 bin k x l r
1416 = Bin (size l + size r + 1) k x l r
1417
1418
1419 {--------------------------------------------------------------------
1420 Eq converts the tree to a list. In a lazy setting, this
1421 actually seems one of the faster methods to compare two trees
1422 and it is certainly the simplest :-)
1423 --------------------------------------------------------------------}
1424 instance (Eq k,Eq a) => Eq (Map k a) where
1425 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
1426
1427 {--------------------------------------------------------------------
1428 Ord
1429 --------------------------------------------------------------------}
1430
1431 instance (Ord k, Ord v) => Ord (Map k v) where
1432 compare m1 m2 = compare (toAscList m1) (toAscList m2)
1433
1434 {--------------------------------------------------------------------
1435 Functor
1436 --------------------------------------------------------------------}
1437 instance Functor (Map k) where
1438 fmap f m = map f m
1439
1440 instance Traversable (Map k) where
1441 traverse f Tip = pure Tip
1442 traverse f (Bin s k v l r)
1443 = flip (Bin s k) <$> traverse f l <*> f v <*> traverse f r
1444
1445 instance Foldable (Map k) where
1446 foldMap _f Tip = mempty
1447 foldMap f (Bin _s _k v l r)
1448 = foldMap f l `mappend` f v `mappend` foldMap f r
1449
1450 {--------------------------------------------------------------------
1451 Read
1452 --------------------------------------------------------------------}
1453 instance (Ord k, Read k, Read e) => Read (Map k e) where
1454 #ifdef __GLASGOW_HASKELL__
1455 readPrec = parens $ prec 10 $ do
1456 Ident "fromList" <- lexP
1457 xs <- readPrec
1458 return (fromList xs)
1459
1460 readListPrec = readListPrecDefault
1461 #else
1462 readsPrec p = readParen (p > 10) $ \ r -> do
1463 ("fromList",s) <- lex r
1464 (xs,t) <- reads s
1465 return (fromList xs,t)
1466 #endif
1467
1468 -- parses a pair of things with the syntax a:=b
1469 readPair :: (Read a, Read b) => ReadS (a,b)
1470 readPair s = do (a, ct1) <- reads s
1471 (":=", ct2) <- lex ct1
1472 (b, ct3) <- reads ct2
1473 return ((a,b), ct3)
1474
1475 {--------------------------------------------------------------------
1476 Show
1477 --------------------------------------------------------------------}
1478 instance (Show k, Show a) => Show (Map k a) where
1479 showsPrec d m = showParen (d > 10) $
1480 showString "fromList " . shows (toList m)
1481
1482 showMap :: (Show k,Show a) => [(k,a)] -> ShowS
1483 showMap []
1484 = showString "{}"
1485 showMap (x:xs)
1486 = showChar '{' . showElem x . showTail xs
1487 where
1488 showTail [] = showChar '}'
1489 showTail (x:xs) = showString ", " . showElem x . showTail xs
1490
1491 showElem (k,x) = shows k . showString " := " . shows x
1492
1493
1494 -- | /O(n)/. Show the tree that implements the map. The tree is shown
1495 -- in a compressed, hanging format.
1496 showTree :: (Show k,Show a) => Map k a -> String
1497 showTree m
1498 = showTreeWith showElem True False m
1499 where
1500 showElem k x = show k ++ ":=" ++ show x
1501
1502
1503 {- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows
1504 the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
1505 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
1506 @wide@ is 'True', an extra wide version is shown.
1507
1508 > Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
1509 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t
1510 > (4,())
1511 > +--(2,())
1512 > | +--(1,())
1513 > | +--(3,())
1514 > +--(5,())
1515 >
1516 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t
1517 > (4,())
1518 > |
1519 > +--(2,())
1520 > | |
1521 > | +--(1,())
1522 > | |
1523 > | +--(3,())
1524 > |
1525 > +--(5,())
1526 >
1527 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t
1528 > +--(5,())
1529 > |
1530 > (4,())
1531 > |
1532 > | +--(3,())
1533 > | |
1534 > +--(2,())
1535 > |
1536 > +--(1,())
1537
1538 -}
1539 showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
1540 showTreeWith showelem hang wide t
1541 | hang = (showsTreeHang showelem wide [] t) ""
1542 | otherwise = (showsTree showelem wide [] [] t) ""
1543
1544 showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
1545 showsTree showelem wide lbars rbars t
1546 = case t of
1547 Tip -> showsBars lbars . showString "|\n"
1548 Bin sz kx x Tip Tip
1549 -> showsBars lbars . showString (showelem kx x) . showString "\n"
1550 Bin sz kx x l r
1551 -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
1552 showWide wide rbars .
1553 showsBars lbars . showString (showelem kx x) . showString "\n" .
1554 showWide wide lbars .
1555 showsTree showelem wide (withEmpty lbars) (withBar lbars) l
1556
1557 showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
1558 showsTreeHang showelem wide bars t
1559 = case t of
1560 Tip -> showsBars bars . showString "|\n"
1561 Bin sz kx x Tip Tip
1562 -> showsBars bars . showString (showelem kx x) . showString "\n"
1563 Bin sz kx x l r
1564 -> showsBars bars . showString (showelem kx x) . showString "\n" .
1565 showWide wide bars .
1566 showsTreeHang showelem wide (withBar bars) l .
1567 showWide wide bars .
1568 showsTreeHang showelem wide (withEmpty bars) r
1569
1570
1571 showWide wide bars
1572 | wide = showString (concat (reverse bars)) . showString "|\n"
1573 | otherwise = id
1574
1575 showsBars :: [String] -> ShowS
1576 showsBars bars
1577 = case bars of
1578 [] -> id
1579 _ -> showString (concat (reverse (tail bars))) . showString node
1580
1581 node = "+--"
1582 withBar bars = "| ":bars
1583 withEmpty bars = " ":bars
1584
1585 {--------------------------------------------------------------------
1586 Typeable
1587 --------------------------------------------------------------------}
1588
1589 #include "Typeable.h"
1590 INSTANCE_TYPEABLE2(Map,mapTc,"Map")
1591
1592 {--------------------------------------------------------------------
1593 Assertions
1594 --------------------------------------------------------------------}
1595 -- | /O(n)/. Test if the internal map structure is valid.
1596 valid :: Ord k => Map k a -> Bool
1597 valid t
1598 = balanced t && ordered t && validsize t
1599
1600 ordered t
1601 = bounded (const True) (const True) t
1602 where
1603 bounded lo hi t
1604 = case t of
1605 Tip -> True
1606 Bin sz kx x l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r
1607
1608 -- | Exported only for "Debug.QuickCheck"
1609 balanced :: Map k a -> Bool
1610 balanced t
1611 = case t of
1612 Tip -> True
1613 Bin sz kx x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
1614 balanced l && balanced r
1615
1616
1617 validsize t
1618 = (realsize t == Just (size t))
1619 where
1620 realsize t
1621 = case t of
1622 Tip -> Just 0
1623 Bin sz kx x l r -> case (realsize l,realsize r) of
1624 (Just n,Just m) | n+m+1 == sz -> Just sz
1625 other -> Nothing
1626
1627 {--------------------------------------------------------------------
1628 Utilities
1629 --------------------------------------------------------------------}
1630 foldlStrict f z xs
1631 = case xs of
1632 [] -> z
1633 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
1634
1635
1636 {-
1637 {--------------------------------------------------------------------
1638 Testing
1639 --------------------------------------------------------------------}
1640 testTree xs = fromList [(x,"*") | x <- xs]
1641 test1 = testTree [1..20]
1642 test2 = testTree [30,29..10]
1643 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
1644
1645 {--------------------------------------------------------------------
1646 QuickCheck
1647 --------------------------------------------------------------------}
1648 qcheck prop
1649 = check config prop
1650 where
1651 config = Config
1652 { configMaxTest = 500
1653 , configMaxFail = 5000
1654 , configSize = \n -> (div n 2 + 3)
1655 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
1656 }
1657
1658
1659 {--------------------------------------------------------------------
1660 Arbitrary, reasonably balanced trees
1661 --------------------------------------------------------------------}
1662 instance (Enum k,Arbitrary a) => Arbitrary (Map k a) where
1663 arbitrary = sized (arbtree 0 maxkey)
1664 where maxkey = 10000
1665
1666 arbtree :: (Enum k,Arbitrary a) => Int -> Int -> Int -> Gen (Map k a)
1667 arbtree lo hi n
1668 | n <= 0 = return Tip
1669 | lo >= hi = return Tip
1670 | otherwise = do{ x <- arbitrary
1671 ; i <- choose (lo,hi)
1672 ; m <- choose (1,30)
1673 ; let (ml,mr) | m==(1::Int)= (1,2)
1674 | m==2 = (2,1)
1675 | m==3 = (1,1)
1676 | otherwise = (2,2)
1677 ; l <- arbtree lo (i-1) (n `div` ml)
1678 ; r <- arbtree (i+1) hi (n `div` mr)
1679 ; return (bin (toEnum i) x l r)
1680 }
1681
1682
1683 {--------------------------------------------------------------------
1684 Valid tree's
1685 --------------------------------------------------------------------}
1686 forValid :: (Show k,Enum k,Show a,Arbitrary a,Testable b) => (Map k a -> b) -> Property
1687 forValid f
1688 = forAll arbitrary $ \t ->
1689 -- classify (balanced t) "balanced" $
1690 classify (size t == 0) "empty" $
1691 classify (size t > 0 && size t <= 10) "small" $
1692 classify (size t > 10 && size t <= 64) "medium" $
1693 classify (size t > 64) "large" $
1694 balanced t ==> f t
1695
1696 forValidIntTree :: Testable a => (Map Int Int -> a) -> Property
1697 forValidIntTree f
1698 = forValid f
1699
1700 forValidUnitTree :: Testable a => (Map Int () -> a) -> Property
1701 forValidUnitTree f
1702 = forValid f
1703
1704
1705 prop_Valid
1706 = forValidUnitTree $ \t -> valid t
1707
1708 {--------------------------------------------------------------------
1709 Single, Insert, Delete
1710 --------------------------------------------------------------------}
1711 prop_Single :: Int -> Int -> Bool
1712 prop_Single k x
1713 = (insert k x empty == singleton k x)
1714
1715 prop_InsertValid :: Int -> Property
1716 prop_InsertValid k
1717 = forValidUnitTree $ \t -> valid (insert k () t)
1718
1719 prop_InsertDelete :: Int -> Map Int () -> Property
1720 prop_InsertDelete k t
1721 = (lookup k t == Nothing) ==> delete k (insert k () t) == t
1722
1723 prop_DeleteValid :: Int -> Property
1724 prop_DeleteValid k
1725 = forValidUnitTree $ \t ->
1726 valid (delete k (insert k () t))
1727
1728 {--------------------------------------------------------------------
1729 Balance
1730 --------------------------------------------------------------------}
1731 prop_Join :: Int -> Property
1732 prop_Join k
1733 = forValidUnitTree $ \t ->
1734 let (l,r) = split k t
1735 in valid (join k () l r)
1736
1737 prop_Merge :: Int -> Property
1738 prop_Merge k
1739 = forValidUnitTree $ \t ->
1740 let (l,r) = split k t
1741 in valid (merge l r)
1742
1743
1744 {--------------------------------------------------------------------
1745 Union
1746 --------------------------------------------------------------------}
1747 prop_UnionValid :: Property
1748 prop_UnionValid
1749 = forValidUnitTree $ \t1 ->
1750 forValidUnitTree $ \t2 ->
1751 valid (union t1 t2)
1752
1753 prop_UnionInsert :: Int -> Int -> Map Int Int -> Bool
1754 prop_UnionInsert k x t
1755 = union (singleton k x) t == insert k x t
1756
1757 prop_UnionAssoc :: Map Int Int -> Map Int Int -> Map Int Int -> Bool
1758 prop_UnionAssoc t1 t2 t3
1759 = union t1 (union t2 t3) == union (union t1 t2) t3
1760
1761 prop_UnionComm :: Map Int Int -> Map Int Int -> Bool
1762 prop_UnionComm t1 t2
1763 = (union t1 t2 == unionWith (\x y -> y) t2 t1)
1764
1765 prop_UnionWithValid
1766 = forValidIntTree $ \t1 ->
1767 forValidIntTree $ \t2 ->
1768 valid (unionWithKey (\k x y -> x+y) t1 t2)
1769
1770 prop_UnionWith :: [(Int,Int)] -> [(Int,Int)] -> Bool
1771 prop_UnionWith xs ys
1772 = sum (elems (unionWith (+) (fromListWith (+) xs) (fromListWith (+) ys)))
1773 == (sum (Prelude.map snd xs) + sum (Prelude.map snd ys))
1774
1775 prop_DiffValid
1776 = forValidUnitTree $ \t1 ->
1777 forValidUnitTree $ \t2 ->
1778 valid (difference t1 t2)
1779
1780 prop_Diff :: [(Int,Int)] -> [(Int,Int)] -> Bool
1781 prop_Diff xs ys
1782 = List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
1783 == List.sort ((List.\\) (nub (Prelude.map fst xs)) (nub (Prelude.map fst ys)))
1784
1785 prop_IntValid
1786 = forValidUnitTree $ \t1 ->
1787 forValidUnitTree $ \t2 ->
1788 valid (intersection t1 t2)
1789
1790 prop_Int :: [(Int,Int)] -> [(Int,Int)] -> Bool
1791 prop_Int xs ys
1792 = List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
1793 == List.sort (nub ((List.intersect) (Prelude.map fst xs) (Prelude.map fst ys)))
1794
1795 {--------------------------------------------------------------------
1796 Lists
1797 --------------------------------------------------------------------}
1798 prop_Ordered
1799 = forAll (choose (5,100)) $ \n ->
1800 let xs = [(x,()) | x <- [0..n::Int]]
1801 in fromAscList xs == fromList xs
1802
1803 prop_List :: [Int] -> Bool
1804 prop_List xs
1805 = (sort (nub xs) == [x | (x,()) <- toList (fromList [(x,()) | x <- xs])])
1806 -}