1 {-# OPTIONS_GHC -fno-bang-patterns #-}
3 -----------------------------------------------------------------------------
6 -- Copyright : (c) Daan Leijen 2002
8 -- Maintainer : libraries@haskell.org
9 -- Stability : provisional
10 -- Portability : portable
12 -- An efficient implementation of maps from keys to values (dictionaries).
14 -- This module is intended to be imported @qualified@, to avoid name
15 -- clashes with Prelude functions. eg.
17 -- > import Data.Map as Map
19 -- The implementation of 'Map' is based on /size balanced/ binary trees (or
20 -- trees of /bounded balance/) as described by:
22 -- * Stephen Adams, \"/Efficient sets: a balancing act/\",
23 -- Journal of Functional Programming 3(4):553-562, October 1993,
24 -- <http://www.swiss.ai.mit.edu/~adams/BB>.
26 -- * J. Nievergelt and E.M. Reingold,
27 -- \"/Binary search trees of bounded balance/\",
28 -- SIAM journal of computing 2(1), March 1973.
30 -- Note that the implementation is /left-biased/ -- the elements of a
31 -- first argument are always preferred to the second, for example in
32 -- 'union' or 'insert'.
33 -----------------------------------------------------------------------------
37 Map -- instance Eq,Show,Read
57 , insertWith, insertWithKey, insertLookupWithKey
118 , fromDistinctAscList
130 , isSubmapOf, isSubmapOfBy
131 , isProperSubmapOf, isProperSubmapOfBy
158 import Prelude hiding (lookup,map,filter,foldr,foldl,null)
159 import qualified Data.Set as Set
160 import qualified Data.List as List
161 import Data.Monoid (Monoid(..))
163 import Control.Applicative (Applicative(..), (<$>))
164 import Data.Traversable (Traversable(traverse))
165 import Data.Foldable (Foldable(foldMap))
169 import qualified Prelude
170 import qualified List
171 import Debug.QuickCheck
172 import List(nub,sort)
175 #if __GLASGOW_HASKELL__
177 import Data.Generics.Basics
178 import Data.Generics.Instances
181 {--------------------------------------------------------------------
183 --------------------------------------------------------------------}
186 -- | /O(log n)/. Find the value at a key.
187 -- Calls 'error' when the element can not be found.
188 (!) :: Ord k => Map k a -> k -> a
191 -- | /O(n+m)/. See 'difference'.
192 (\\) :: Ord k => Map k a -> Map k b -> Map k a
193 m1 \\ m2 = difference m1 m2
195 {--------------------------------------------------------------------
197 --------------------------------------------------------------------}
198 -- | A Map from keys @k@ to values @a@.
200 | Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
204 instance (Ord k) => Monoid (Map k v) where
209 #if __GLASGOW_HASKELL__
211 {--------------------------------------------------------------------
213 --------------------------------------------------------------------}
215 -- This instance preserves data abstraction at the cost of inefficiency.
216 -- We omit reflection services for the sake of data abstraction.
218 instance (Data k, Data a, Ord k) => Data (Map k a) where
219 gfoldl f z map = z fromList `f` (toList map)
220 toConstr _ = error "toConstr"
221 gunfold _ _ = error "gunfold"
222 dataTypeOf _ = mkNorepType "Data.Map.Map"
223 dataCast2 f = gcast2 f
227 {--------------------------------------------------------------------
229 --------------------------------------------------------------------}
230 -- | /O(1)/. Is the map empty?
231 null :: Map k a -> Bool
235 Bin sz k x l r -> False
237 -- | /O(1)/. The number of elements in the map.
238 size :: Map k a -> Int
245 -- | /O(log n)/. Lookup the value at a key in the map.
246 lookup :: (Monad m,Ord k) => k -> Map k a -> m a
247 lookup k t = case lookup' k t of
249 Nothing -> fail "Data.Map.lookup: Key not found"
250 lookup' :: Ord k => k -> Map k a -> Maybe a
255 -> case compare k kx of
260 lookupAssoc :: Ord k => k -> Map k a -> Maybe (k,a)
265 -> case compare k kx of
266 LT -> lookupAssoc k l
267 GT -> lookupAssoc k r
270 -- | /O(log n)/. Is the key a member of the map?
271 member :: Ord k => k -> Map k a -> Bool
277 -- | /O(log n)/. Is the key not a member of the map?
278 notMember :: Ord k => k -> Map k a -> Bool
279 notMember k m = not $ member k m
281 -- | /O(log n)/. Find the value at a key.
282 -- Calls 'error' when the element can not be found.
283 find :: Ord k => k -> Map k a -> a
286 Nothing -> error "Map.find: element not in the map"
289 -- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns
290 -- the value at key @k@ or returns @def@ when the key is not in the map.
291 findWithDefault :: Ord k => a -> k -> Map k a -> a
292 findWithDefault def k m
299 {--------------------------------------------------------------------
301 --------------------------------------------------------------------}
302 -- | /O(1)/. The empty map.
307 -- | /O(1)/. A map with a single element.
308 singleton :: k -> a -> Map k a
312 {--------------------------------------------------------------------
314 --------------------------------------------------------------------}
315 -- | /O(log n)/. Insert a new key and value in the map.
316 -- If the key is already present in the map, the associated value is
317 -- replaced with the supplied value, i.e. 'insert' is equivalent to
318 -- @'insertWith' 'const'@.
319 insert :: Ord k => k -> a -> Map k a -> Map k a
322 Tip -> singleton kx x
324 -> case compare kx ky of
325 LT -> balance ky y (insert kx x l) r
326 GT -> balance ky y l (insert kx x r)
327 EQ -> Bin sz kx x l r
329 -- | /O(log n)/. Insert with a combining function.
330 -- @'insertWith' f key value mp@
331 -- will insert the pair (key, value) into @mp@ if key does
332 -- not exist in the map. If the key does exist, the function will
333 -- insert the pair @(key, f new_value old_value)@.
334 insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
336 = insertWithKey (\k x y -> f x y) k x m
338 -- | /O(log n)/. Insert with a combining function.
339 -- @'insertWithKey' f key value mp@
340 -- will insert the pair (key, value) into @mp@ if key does
341 -- not exist in the map. If the key does exist, the function will
342 -- insert the pair @(key,f key new_value old_value)@.
343 -- Note that the key passed to f is the same key passed to 'insertWithKey'.
344 insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
345 insertWithKey f kx x t
347 Tip -> singleton kx x
349 -> case compare kx ky of
350 LT -> balance ky y (insertWithKey f kx x l) r
351 GT -> balance ky y l (insertWithKey f kx x r)
352 EQ -> Bin sy kx (f kx x y) l r
354 -- | /O(log n)/. The expression (@'insertLookupWithKey' f k x map@)
355 -- is a pair where the first element is equal to (@'lookup' k map@)
356 -- and the second element equal to (@'insertWithKey' f k x map@).
357 insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a,Map k a)
358 insertLookupWithKey f kx x t
360 Tip -> (Nothing, singleton kx x)
362 -> case compare kx ky of
363 LT -> let (found,l') = insertLookupWithKey f kx x l in (found,balance ky y l' r)
364 GT -> let (found,r') = insertLookupWithKey f kx x r in (found,balance ky y l r')
365 EQ -> (Just y, Bin sy kx (f kx x y) l r)
367 {--------------------------------------------------------------------
369 [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
370 --------------------------------------------------------------------}
371 -- | /O(log n)/. Delete a key and its value from the map. When the key is not
372 -- a member of the map, the original map is returned.
373 delete :: Ord k => k -> Map k a -> Map k a
378 -> case compare k kx of
379 LT -> balance kx x (delete k l) r
380 GT -> balance kx x l (delete k r)
383 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
384 -- a member of the map, the original map is returned.
385 adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
387 = adjustWithKey (\k x -> f x) k m
389 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
390 -- a member of the map, the original map is returned.
391 adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
393 = updateWithKey (\k x -> Just (f k x)) k m
395 -- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@
396 -- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
397 -- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
398 update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
400 = updateWithKey (\k x -> f x) k m
402 -- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the
403 -- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',
404 -- the element is deleted. If it is (@'Just' y@), the key @k@ is bound
405 -- to the new value @y@.
406 updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
411 -> case compare k kx of
412 LT -> balance kx x (updateWithKey f k l) r
413 GT -> balance kx x l (updateWithKey f k r)
415 Just x' -> Bin sx kx x' l r
418 -- | /O(log n)/. Lookup and update.
419 updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
420 updateLookupWithKey f k t
424 -> case compare k kx of
425 LT -> let (found,l') = updateLookupWithKey f k l in (found,balance kx x l' r)
426 GT -> let (found,r') = updateLookupWithKey f k r in (found,balance kx x l r')
428 Just x' -> (Just x',Bin sx kx x' l r)
429 Nothing -> (Just x,glue l r)
431 -- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
432 -- 'alter' can be used to insert, delete, or update a value in a 'Map'.
433 -- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@
434 alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
437 Tip -> case f Nothing of
439 Just x -> singleton k x
441 -> case compare k kx of
442 LT -> balance kx x (alter f k l) r
443 GT -> balance kx x l (alter f k r)
444 EQ -> case f (Just x) of
445 Just x' -> Bin sx kx x' l r
448 {--------------------------------------------------------------------
450 --------------------------------------------------------------------}
451 -- | /O(log n)/. Return the /index/ of a key. The index is a number from
452 -- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
453 -- the key is not a 'member' of the map.
454 findIndex :: Ord k => k -> Map k a -> Int
456 = case lookupIndex k t of
457 Nothing -> error "Map.findIndex: element is not in the map"
460 -- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
461 -- /0/ up to, but not including, the 'size' of the map.
462 lookupIndex :: (Monad m,Ord k) => k -> Map k a -> m Int
463 lookupIndex k t = case lookup 0 t of
464 Nothing -> fail "Data.Map.lookupIndex: Key not found."
467 lookup idx Tip = Nothing
468 lookup idx (Bin _ kx x l r)
469 = case compare k kx of
471 GT -> lookup (idx + size l + 1) r
472 EQ -> Just (idx + size l)
474 -- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
475 -- invalid index is used.
476 elemAt :: Int -> Map k a -> (k,a)
477 elemAt i Tip = error "Map.elemAt: index out of range"
478 elemAt i (Bin _ kx x l r)
479 = case compare i sizeL of
481 GT -> elemAt (i-sizeL-1) r
486 -- | /O(log n)/. Update the element at /index/. Calls 'error' when an
487 -- invalid index is used.
488 updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
489 updateAt f i Tip = error "Map.updateAt: index out of range"
490 updateAt f i (Bin sx kx x l r)
491 = case compare i sizeL of
493 GT -> updateAt f (i-sizeL-1) r
495 Just x' -> Bin sx kx x' l r
500 -- | /O(log n)/. Delete the element at /index/.
501 -- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).
502 deleteAt :: Int -> Map k a -> Map k a
504 = updateAt (\k x -> Nothing) i map
507 {--------------------------------------------------------------------
509 --------------------------------------------------------------------}
510 -- | /O(log n)/. The minimal key of the map.
511 findMin :: Map k a -> (k,a)
512 findMin (Bin _ kx x Tip r) = (kx,x)
513 findMin (Bin _ kx x l r) = findMin l
514 findMin Tip = error "Map.findMin: empty tree has no minimal element"
516 -- | /O(log n)/. The maximal key of the map.
517 findMax :: Map k a -> (k,a)
518 findMax (Bin _ kx x l Tip) = (kx,x)
519 findMax (Bin _ kx x l r) = findMax r
520 findMax Tip = error "Map.findMax: empty tree has no maximal element"
522 -- | /O(log n)/. Delete the minimal key.
523 deleteMin :: Map k a -> Map k a
524 deleteMin (Bin _ kx x Tip r) = r
525 deleteMin (Bin _ kx x l r) = balance kx x (deleteMin l) r
528 -- | /O(log n)/. Delete the maximal key.
529 deleteMax :: Map k a -> Map k a
530 deleteMax (Bin _ kx x l Tip) = l
531 deleteMax (Bin _ kx x l r) = balance kx x l (deleteMax r)
534 -- | /O(log n)/. Update the value at the minimal key.
535 updateMin :: (a -> Maybe a) -> Map k a -> Map k a
537 = updateMinWithKey (\k x -> f x) m
539 -- | /O(log n)/. Update the value at the maximal key.
540 updateMax :: (a -> Maybe a) -> Map k a -> Map k a
542 = updateMaxWithKey (\k x -> f x) m
545 -- | /O(log n)/. Update the value at the minimal key.
546 updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
549 Bin sx kx x Tip r -> case f kx x of
551 Just x' -> Bin sx kx x' Tip r
552 Bin sx kx x l r -> balance kx x (updateMinWithKey f l) r
555 -- | /O(log n)/. Update the value at the maximal key.
556 updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
559 Bin sx kx x l Tip -> case f kx x of
561 Just x' -> Bin sx kx x' l Tip
562 Bin sx kx x l r -> balance kx x l (updateMaxWithKey f r)
566 {--------------------------------------------------------------------
568 --------------------------------------------------------------------}
569 -- | The union of a list of maps:
570 -- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).
571 unions :: Ord k => [Map k a] -> Map k a
573 = foldlStrict union empty ts
575 -- | The union of a list of maps, with a combining operation:
576 -- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).
577 unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a
579 = foldlStrict (unionWith f) empty ts
582 -- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.
583 -- It prefers @t1@ when duplicate keys are encountered,
584 -- i.e. (@'union' == 'unionWith' 'const'@).
585 -- The implementation uses the efficient /hedge-union/ algorithm.
586 -- Hedge-union is more efficient on (bigset `union` smallset)
587 union :: Ord k => Map k a -> Map k a -> Map k a
590 union t1 t2 = hedgeUnionL (const LT) (const GT) t1 t2
592 -- left-biased hedge union
593 hedgeUnionL cmplo cmphi t1 Tip
595 hedgeUnionL cmplo cmphi Tip (Bin _ kx x l r)
596 = join kx x (filterGt cmplo l) (filterLt cmphi r)
597 hedgeUnionL cmplo cmphi (Bin _ kx x l r) t2
598 = join kx x (hedgeUnionL cmplo cmpkx l (trim cmplo cmpkx t2))
599 (hedgeUnionL cmpkx cmphi r (trim cmpkx cmphi t2))
601 cmpkx k = compare kx k
603 -- right-biased hedge union
604 hedgeUnionR cmplo cmphi t1 Tip
606 hedgeUnionR cmplo cmphi Tip (Bin _ kx x l r)
607 = join kx x (filterGt cmplo l) (filterLt cmphi r)
608 hedgeUnionR cmplo cmphi (Bin _ kx x l r) t2
609 = join kx newx (hedgeUnionR cmplo cmpkx l lt)
610 (hedgeUnionR cmpkx cmphi r gt)
612 cmpkx k = compare kx k
613 lt = trim cmplo cmpkx t2
614 (found,gt) = trimLookupLo kx cmphi t2
619 {--------------------------------------------------------------------
620 Union with a combining function
621 --------------------------------------------------------------------}
622 -- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
623 unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
625 = unionWithKey (\k x y -> f x y) m1 m2
628 -- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
629 -- Hedge-union is more efficient on (bigset `union` smallset).
630 unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
631 unionWithKey f Tip t2 = t2
632 unionWithKey f t1 Tip = t1
633 unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
635 hedgeUnionWithKey f cmplo cmphi t1 Tip
637 hedgeUnionWithKey f cmplo cmphi Tip (Bin _ kx x l r)
638 = join kx x (filterGt cmplo l) (filterLt cmphi r)
639 hedgeUnionWithKey f cmplo cmphi (Bin _ kx x l r) t2
640 = join kx newx (hedgeUnionWithKey f cmplo cmpkx l lt)
641 (hedgeUnionWithKey f cmpkx cmphi r gt)
643 cmpkx k = compare kx k
644 lt = trim cmplo cmpkx t2
645 (found,gt) = trimLookupLo kx cmphi t2
648 Just (_,y) -> f kx x y
650 {--------------------------------------------------------------------
652 --------------------------------------------------------------------}
653 -- | /O(n+m)/. Difference of two maps.
654 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
655 difference :: Ord k => Map k a -> Map k b -> Map k a
656 difference Tip t2 = Tip
657 difference t1 Tip = t1
658 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
660 hedgeDiff cmplo cmphi Tip t
662 hedgeDiff cmplo cmphi (Bin _ kx x l r) Tip
663 = join kx x (filterGt cmplo l) (filterLt cmphi r)
664 hedgeDiff cmplo cmphi t (Bin _ kx x l r)
665 = merge (hedgeDiff cmplo cmpkx (trim cmplo cmpkx t) l)
666 (hedgeDiff cmpkx cmphi (trim cmpkx cmphi t) r)
668 cmpkx k = compare kx k
670 -- | /O(n+m)/. Difference with a combining function.
671 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
672 differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
673 differenceWith f m1 m2
674 = differenceWithKey (\k x y -> f x y) m1 m2
676 -- | /O(n+m)/. Difference with a combining function. When two equal keys are
677 -- encountered, the combining function is applied to the key and both values.
678 -- If it returns 'Nothing', the element is discarded (proper set difference). If
679 -- it returns (@'Just' y@), the element is updated with a new value @y@.
680 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
681 differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
682 differenceWithKey f Tip t2 = Tip
683 differenceWithKey f t1 Tip = t1
684 differenceWithKey f t1 t2 = hedgeDiffWithKey f (const LT) (const GT) t1 t2
686 hedgeDiffWithKey f cmplo cmphi Tip t
688 hedgeDiffWithKey f cmplo cmphi (Bin _ kx x l r) Tip
689 = join kx x (filterGt cmplo l) (filterLt cmphi r)
690 hedgeDiffWithKey f cmplo cmphi t (Bin _ kx x l r)
692 Nothing -> merge tl tr
695 Nothing -> merge tl tr
696 Just z -> join ky z tl tr
698 cmpkx k = compare kx k
699 lt = trim cmplo cmpkx t
700 (found,gt) = trimLookupLo kx cmphi t
701 tl = hedgeDiffWithKey f cmplo cmpkx lt l
702 tr = hedgeDiffWithKey f cmpkx cmphi gt r
706 {--------------------------------------------------------------------
708 --------------------------------------------------------------------}
709 -- | /O(n+m)/. Intersection of two maps. The values in the first
710 -- map are returned, i.e. (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).
711 intersection :: Ord k => Map k a -> Map k b -> Map k a
713 = intersectionWithKey (\k x y -> x) m1 m2
715 -- | /O(n+m)/. Intersection with a combining function.
716 intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
717 intersectionWith f m1 m2
718 = intersectionWithKey (\k x y -> f x y) m1 m2
720 -- | /O(n+m)/. Intersection with a combining function.
721 -- Intersection is more efficient on (bigset `intersection` smallset)
722 --intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
723 --intersectionWithKey f Tip t = Tip
724 --intersectionWithKey f t Tip = Tip
725 --intersectionWithKey f t1 t2 = intersectWithKey f t1 t2
727 --intersectWithKey f Tip t = Tip
728 --intersectWithKey f t Tip = Tip
729 --intersectWithKey f t (Bin _ kx x l r)
731 -- Nothing -> merge tl tr
732 -- Just y -> join kx (f kx y x) tl tr
734 -- (lt,found,gt) = splitLookup kx t
735 -- tl = intersectWithKey f lt l
736 -- tr = intersectWithKey f gt r
739 intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
740 intersectionWithKey f Tip t = Tip
741 intersectionWithKey f t Tip = Tip
742 intersectionWithKey f t1@(Bin s1 k1 x1 l1 r1) t2@(Bin s2 k2 x2 l2 r2) =
744 let (lt,found,gt) = splitLookupWithKey k2 t1
745 tl = intersectionWithKey f lt l2
746 tr = intersectionWithKey f gt r2
748 Just (k,x) -> join k (f k x x2) tl tr
749 Nothing -> merge tl tr
750 else let (lt,found,gt) = splitLookup k1 t2
751 tl = intersectionWithKey f l1 lt
752 tr = intersectionWithKey f r1 gt
754 Just x -> join k1 (f k1 x1 x) tl tr
755 Nothing -> merge tl tr
759 {--------------------------------------------------------------------
761 --------------------------------------------------------------------}
763 -- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
764 isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
766 = isSubmapOfBy (==) m1 m2
769 The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if
770 all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when
771 applied to their respective values. For example, the following
772 expressions are all 'True':
774 > isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
775 > isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
776 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
778 But the following are all 'False':
780 > isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
781 > isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
782 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
784 isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool
786 = (size t1 <= size t2) && (submap' f t1 t2)
788 submap' f Tip t = True
789 submap' f t Tip = False
790 submap' f (Bin _ kx x l r) t
793 Just y -> f x y && submap' f l lt && submap' f r gt
795 (lt,found,gt) = splitLookup kx t
797 -- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
798 -- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
799 isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
800 isProperSubmapOf m1 m2
801 = isProperSubmapOfBy (==) m1 m2
803 {- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
804 The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
805 @m1@ and @m2@ are not equal,
806 all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
807 applied to their respective values. For example, the following
808 expressions are all 'True':
810 > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
811 > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
813 But the following are all 'False':
815 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
816 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
817 > isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
819 isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
820 isProperSubmapOfBy f t1 t2
821 = (size t1 < size t2) && (submap' f t1 t2)
823 {--------------------------------------------------------------------
825 --------------------------------------------------------------------}
826 -- | /O(n)/. Filter all values that satisfy the predicate.
827 filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
829 = filterWithKey (\k x -> p x) m
831 -- | /O(n)/. Filter all keys\/values that satisfy the predicate.
832 filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
833 filterWithKey p Tip = Tip
834 filterWithKey p (Bin _ kx x l r)
835 | p kx x = join kx x (filterWithKey p l) (filterWithKey p r)
836 | otherwise = merge (filterWithKey p l) (filterWithKey p r)
839 -- | /O(n)/. partition the map according to a predicate. The first
840 -- map contains all elements that satisfy the predicate, the second all
841 -- elements that fail the predicate. See also 'split'.
842 partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a)
844 = partitionWithKey (\k x -> p x) m
846 -- | /O(n)/. partition the map according to a predicate. The first
847 -- map contains all elements that satisfy the predicate, the second all
848 -- elements that fail the predicate. See also 'split'.
849 partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
850 partitionWithKey p Tip = (Tip,Tip)
851 partitionWithKey p (Bin _ kx x l r)
852 | p kx x = (join kx x l1 r1,merge l2 r2)
853 | otherwise = (merge l1 r1,join kx x l2 r2)
855 (l1,l2) = partitionWithKey p l
856 (r1,r2) = partitionWithKey p r
859 {--------------------------------------------------------------------
861 --------------------------------------------------------------------}
862 -- | /O(n)/. Map a function over all values in the map.
863 map :: (a -> b) -> Map k a -> Map k b
865 = mapWithKey (\k x -> f x) m
867 -- | /O(n)/. Map a function over all values in the map.
868 mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
869 mapWithKey f Tip = Tip
870 mapWithKey f (Bin sx kx x l r)
871 = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)
873 -- | /O(n)/. The function 'mapAccum' threads an accumulating
874 -- argument through the map in ascending order of keys.
875 mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
877 = mapAccumWithKey (\a k x -> f a x) a m
879 -- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating
880 -- argument through the map in ascending order of keys.
881 mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
882 mapAccumWithKey f a t
885 -- | /O(n)/. The function 'mapAccumL' threads an accumulating
886 -- argument throught the map in ascending order of keys.
887 mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
892 -> let (a1,l') = mapAccumL f a l
894 (a3,r') = mapAccumL f a2 r
895 in (a3,Bin sx kx x' l' r')
897 -- | /O(n)/. The function 'mapAccumR' threads an accumulating
898 -- argument throught the map in descending order of keys.
899 mapAccumR :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
904 -> let (a1,r') = mapAccumR f a r
906 (a3,l') = mapAccumR f a2 l
907 in (a3,Bin sx kx x' l' r')
910 -- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
912 -- The size of the result may be smaller if @f@ maps two or more distinct
913 -- keys to the same new key. In this case the value at the smallest of
914 -- these keys is retained.
916 mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
917 mapKeys = mapKeysWith (\x y->x)
920 -- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
922 -- The size of the result may be smaller if @f@ maps two or more distinct
923 -- keys to the same new key. In this case the associated values will be
924 -- combined using @c@.
926 mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a
927 mapKeysWith c f = fromListWith c . List.map fFirst . toList
928 where fFirst (x,y) = (f x, y)
932 -- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
933 -- is strictly monotonic.
934 -- /The precondition is not checked./
935 -- Semi-formally, we have:
937 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
938 -- > ==> mapKeysMonotonic f s == mapKeys f s
939 -- > where ls = keys s
941 mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a
942 mapKeysMonotonic f Tip = Tip
943 mapKeysMonotonic f (Bin sz k x l r) =
944 Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)
946 {--------------------------------------------------------------------
948 --------------------------------------------------------------------}
950 -- | /O(n)/. Fold the values in the map, such that
951 -- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.
954 -- > elems map = fold (:) [] map
956 fold :: (a -> b -> b) -> b -> Map k a -> b
958 = foldWithKey (\k x z -> f x z) z m
960 -- | /O(n)/. Fold the keys and values in the map, such that
961 -- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
964 -- > keys map = foldWithKey (\k x ks -> k:ks) [] map
966 foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
970 -- | /O(n)/. In-order fold.
971 foldi :: (k -> a -> b -> b -> b) -> b -> Map k a -> b
973 foldi f z (Bin _ kx x l r) = f kx x (foldi f z l) (foldi f z r)
975 -- | /O(n)/. Post-order fold.
976 foldr :: (k -> a -> b -> b) -> b -> Map k a -> b
978 foldr f z (Bin _ kx x l r) = foldr f (f kx x (foldr f z r)) l
980 -- | /O(n)/. Pre-order fold.
981 foldl :: (b -> k -> a -> b) -> b -> Map k a -> b
983 foldl f z (Bin _ kx x l r) = foldl f (f (foldl f z l) kx x) r
985 {--------------------------------------------------------------------
987 --------------------------------------------------------------------}
989 -- Return all elements of the map in the ascending order of their keys.
990 elems :: Map k a -> [a]
992 = [x | (k,x) <- assocs m]
994 -- | /O(n)/. Return all keys of the map in ascending order.
995 keys :: Map k a -> [k]
997 = [k | (k,x) <- assocs m]
999 -- | /O(n)/. The set of all keys of the map.
1000 keysSet :: Map k a -> Set.Set k
1001 keysSet m = Set.fromDistinctAscList (keys m)
1003 -- | /O(n)/. Return all key\/value pairs in the map in ascending key order.
1004 assocs :: Map k a -> [(k,a)]
1008 {--------------------------------------------------------------------
1010 use [foldlStrict] to reduce demand on the control-stack
1011 --------------------------------------------------------------------}
1012 -- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
1013 fromList :: Ord k => [(k,a)] -> Map k a
1015 = foldlStrict ins empty xs
1017 ins t (k,x) = insert k x t
1019 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
1020 fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a
1022 = fromListWithKey (\k x y -> f x y) xs
1024 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
1025 fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1026 fromListWithKey f xs
1027 = foldlStrict ins empty xs
1029 ins t (k,x) = insertWithKey f k x t
1031 -- | /O(n)/. Convert to a list of key\/value pairs.
1032 toList :: Map k a -> [(k,a)]
1033 toList t = toAscList t
1035 -- | /O(n)/. Convert to an ascending list.
1036 toAscList :: Map k a -> [(k,a)]
1037 toAscList t = foldr (\k x xs -> (k,x):xs) [] t
1040 toDescList :: Map k a -> [(k,a)]
1041 toDescList t = foldl (\xs k x -> (k,x):xs) [] t
1044 {--------------------------------------------------------------------
1045 Building trees from ascending/descending lists can be done in linear time.
1047 Note that if [xs] is ascending that:
1048 fromAscList xs == fromList xs
1049 fromAscListWith f xs == fromListWith f xs
1050 --------------------------------------------------------------------}
1051 -- | /O(n)/. Build a map from an ascending list in linear time.
1052 -- /The precondition (input list is ascending) is not checked./
1053 fromAscList :: Eq k => [(k,a)] -> Map k a
1055 = fromAscListWithKey (\k x y -> x) xs
1057 -- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
1058 -- /The precondition (input list is ascending) is not checked./
1059 fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
1060 fromAscListWith f xs
1061 = fromAscListWithKey (\k x y -> f x y) xs
1063 -- | /O(n)/. Build a map from an ascending list in linear time with a
1064 -- combining function for equal keys.
1065 -- /The precondition (input list is ascending) is not checked./
1066 fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1067 fromAscListWithKey f xs
1068 = fromDistinctAscList (combineEq f xs)
1070 -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
1075 (x:xx) -> combineEq' x xx
1077 combineEq' z [] = [z]
1078 combineEq' z@(kz,zz) (x@(kx,xx):xs)
1079 | kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs
1080 | otherwise = z:combineEq' x xs
1083 -- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
1084 -- /The precondition is not checked./
1085 fromDistinctAscList :: [(k,a)] -> Map k a
1086 fromDistinctAscList xs
1087 = build const (length xs) xs
1089 -- 1) use continutations so that we use heap space instead of stack space.
1090 -- 2) special case for n==5 to build bushier trees.
1091 build c 0 xs = c Tip xs
1092 build c 5 xs = case xs of
1093 ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
1094 -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
1095 build c n xs = seq nr $ build (buildR nr c) nl xs
1100 buildR n c l ((k,x):ys) = build (buildB l k x c) n ys
1101 buildB l k x c r zs = c (bin k x l r) zs
1105 {--------------------------------------------------------------------
1106 Utility functions that return sub-ranges of the original
1107 tree. Some functions take a comparison function as argument to
1108 allow comparisons against infinite values. A function [cmplo k]
1109 should be read as [compare lo k].
1111 [trim cmplo cmphi t] A tree that is either empty or where [cmplo k == LT]
1112 and [cmphi k == GT] for the key [k] of the root.
1113 [filterGt cmp t] A tree where for all keys [k]. [cmp k == LT]
1114 [filterLt cmp t] A tree where for all keys [k]. [cmp k == GT]
1116 [split k t] Returns two trees [l] and [r] where all keys
1117 in [l] are <[k] and all keys in [r] are >[k].
1118 [splitLookup k t] Just like [split] but also returns whether [k]
1119 was found in the tree.
1120 --------------------------------------------------------------------}
1122 {--------------------------------------------------------------------
1123 [trim lo hi t] trims away all subtrees that surely contain no
1124 values between the range [lo] to [hi]. The returned tree is either
1125 empty or the key of the root is between @lo@ and @hi@.
1126 --------------------------------------------------------------------}
1127 trim :: (k -> Ordering) -> (k -> Ordering) -> Map k a -> Map k a
1128 trim cmplo cmphi Tip = Tip
1129 trim cmplo cmphi t@(Bin sx kx x l r)
1131 LT -> case cmphi kx of
1133 le -> trim cmplo cmphi l
1134 ge -> trim cmplo cmphi r
1136 trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a)
1137 trimLookupLo lo cmphi Tip = (Nothing,Tip)
1138 trimLookupLo lo cmphi t@(Bin sx kx x l r)
1139 = case compare lo kx of
1140 LT -> case cmphi kx of
1141 GT -> (lookupAssoc lo t, t)
1142 le -> trimLookupLo lo cmphi l
1143 GT -> trimLookupLo lo cmphi r
1144 EQ -> (Just (kx,x),trim (compare lo) cmphi r)
1147 {--------------------------------------------------------------------
1148 [filterGt k t] filter all keys >[k] from tree [t]
1149 [filterLt k t] filter all keys <[k] from tree [t]
1150 --------------------------------------------------------------------}
1151 filterGt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1152 filterGt cmp Tip = Tip
1153 filterGt cmp (Bin sx kx x l r)
1155 LT -> join kx x (filterGt cmp l) r
1156 GT -> filterGt cmp r
1159 filterLt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1160 filterLt cmp Tip = Tip
1161 filterLt cmp (Bin sx kx x l r)
1163 LT -> filterLt cmp l
1164 GT -> join kx x l (filterLt cmp r)
1167 {--------------------------------------------------------------------
1169 --------------------------------------------------------------------}
1170 -- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where
1171 -- 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@.
1172 split :: Ord k => k -> Map k a -> (Map k a,Map k a)
1173 split k Tip = (Tip,Tip)
1174 split k (Bin sx kx x l r)
1175 = case compare k kx of
1176 LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
1177 GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
1180 -- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just
1181 -- like 'split' but also returns @'lookup' k map@.
1182 splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)
1183 splitLookup k Tip = (Tip,Nothing,Tip)
1184 splitLookup k (Bin sx kx x l r)
1185 = case compare k kx of
1186 LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
1187 GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
1191 splitLookupWithKey :: Ord k => k -> Map k a -> (Map k a,Maybe (k,a),Map k a)
1192 splitLookupWithKey k Tip = (Tip,Nothing,Tip)
1193 splitLookupWithKey k (Bin sx kx x l r)
1194 = case compare k kx of
1195 LT -> let (lt,z,gt) = splitLookupWithKey k l in (lt,z,join kx x gt r)
1196 GT -> let (lt,z,gt) = splitLookupWithKey k r in (join kx x l lt,z,gt)
1197 EQ -> (l,Just (kx, x),r)
1199 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
1200 -- element was found in the original set.
1201 splitMember :: Ord k => k -> Map k a -> (Map k a,Bool,Map k a)
1202 splitMember x t = let (l,m,r) = splitLookup x t in
1203 (l,maybe False (const True) m,r)
1206 {--------------------------------------------------------------------
1207 Utility functions that maintain the balance properties of the tree.
1208 All constructors assume that all values in [l] < [k] and all values
1209 in [r] > [k], and that [l] and [r] are valid trees.
1211 In order of sophistication:
1212 [Bin sz k x l r] The type constructor.
1213 [bin k x l r] Maintains the correct size, assumes that both [l]
1214 and [r] are balanced with respect to each other.
1215 [balance k x l r] Restores the balance and size.
1216 Assumes that the original tree was balanced and
1217 that [l] or [r] has changed by at most one element.
1218 [join k x l r] Restores balance and size.
1220 Furthermore, we can construct a new tree from two trees. Both operations
1221 assume that all values in [l] < all values in [r] and that [l] and [r]
1223 [glue l r] Glues [l] and [r] together. Assumes that [l] and
1224 [r] are already balanced with respect to each other.
1225 [merge l r] Merges two trees and restores balance.
1227 Note: in contrast to Adam's paper, we use (<=) comparisons instead
1228 of (<) comparisons in [join], [merge] and [balance].
1229 Quickcheck (on [difference]) showed that this was necessary in order
1230 to maintain the invariants. It is quite unsatisfactory that I haven't
1231 been able to find out why this is actually the case! Fortunately, it
1232 doesn't hurt to be a bit more conservative.
1233 --------------------------------------------------------------------}
1235 {--------------------------------------------------------------------
1237 --------------------------------------------------------------------}
1238 join :: Ord k => k -> a -> Map k a -> Map k a -> Map k a
1239 join kx x Tip r = insertMin kx x r
1240 join kx x l Tip = insertMax kx x l
1241 join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
1242 | delta*sizeL <= sizeR = balance kz z (join kx x l lz) rz
1243 | delta*sizeR <= sizeL = balance ky y ly (join kx x ry r)
1244 | otherwise = bin kx x l r
1247 -- insertMin and insertMax don't perform potentially expensive comparisons.
1248 insertMax,insertMin :: k -> a -> Map k a -> Map k a
1251 Tip -> singleton kx x
1253 -> balance ky y l (insertMax kx x r)
1257 Tip -> singleton kx x
1259 -> balance ky y (insertMin kx x l) r
1261 {--------------------------------------------------------------------
1262 [merge l r]: merges two trees.
1263 --------------------------------------------------------------------}
1264 merge :: Map k a -> Map k a -> Map k a
1267 merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
1268 | delta*sizeL <= sizeR = balance ky y (merge l ly) ry
1269 | delta*sizeR <= sizeL = balance kx x lx (merge rx r)
1270 | otherwise = glue l r
1272 {--------------------------------------------------------------------
1273 [glue l r]: glues two trees together.
1274 Assumes that [l] and [r] are already balanced with respect to each other.
1275 --------------------------------------------------------------------}
1276 glue :: Map k a -> Map k a -> Map k a
1280 | size l > size r = let ((km,m),l') = deleteFindMax l in balance km m l' r
1281 | otherwise = let ((km,m),r') = deleteFindMin r in balance km m l r'
1284 -- | /O(log n)/. Delete and find the minimal element.
1285 deleteFindMin :: Map k a -> ((k,a),Map k a)
1288 Bin _ k x Tip r -> ((k,x),r)
1289 Bin _ k x l r -> let (km,l') = deleteFindMin l in (km,balance k x l' r)
1290 Tip -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)
1292 -- | /O(log n)/. Delete and find the maximal element.
1293 deleteFindMax :: Map k a -> ((k,a),Map k a)
1296 Bin _ k x l Tip -> ((k,x),l)
1297 Bin _ k x l r -> let (km,r') = deleteFindMax r in (km,balance k x l r')
1298 Tip -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
1301 {--------------------------------------------------------------------
1302 [balance l x r] balances two trees with value x.
1303 The sizes of the trees should balance after decreasing the
1304 size of one of them. (a rotation).
1306 [delta] is the maximal relative difference between the sizes of
1307 two trees, it corresponds with the [w] in Adams' paper.
1308 [ratio] is the ratio between an outer and inner sibling of the
1309 heavier subtree in an unbalanced setting. It determines
1310 whether a double or single rotation should be performed
1311 to restore balance. It is correspondes with the inverse
1312 of $\alpha$ in Adam's article.
1315 - [delta] should be larger than 4.646 with a [ratio] of 2.
1316 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
1318 - A lower [delta] leads to a more 'perfectly' balanced tree.
1319 - A higher [delta] performs less rebalancing.
1321 - Balancing is automatic for random data and a balancing
1322 scheme is only necessary to avoid pathological worst cases.
1323 Almost any choice will do, and in practice, a rather large
1324 [delta] may perform better than smaller one.
1326 Note: in contrast to Adam's paper, we use a ratio of (at least) [2]
1327 to decide whether a single or double rotation is needed. Allthough
1328 he actually proves that this ratio is needed to maintain the
1329 invariants, his implementation uses an invalid ratio of [1].
1330 --------------------------------------------------------------------}
1335 balance :: k -> a -> Map k a -> Map k a -> Map k a
1337 | sizeL + sizeR <= 1 = Bin sizeX k x l r
1338 | sizeR >= delta*sizeL = rotateL k x l r
1339 | sizeL >= delta*sizeR = rotateR k x l r
1340 | otherwise = Bin sizeX k x l r
1344 sizeX = sizeL + sizeR + 1
1347 rotateL k x l r@(Bin _ _ _ ly ry)
1348 | size ly < ratio*size ry = singleL k x l r
1349 | otherwise = doubleL k x l r
1351 rotateR k x l@(Bin _ _ _ ly ry) r
1352 | size ry < ratio*size ly = singleR k x l r
1353 | otherwise = doubleR k x l r
1356 singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
1357 singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)
1359 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)
1360 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)
1363 {--------------------------------------------------------------------
1364 The bin constructor maintains the size of the tree
1365 --------------------------------------------------------------------}
1366 bin :: k -> a -> Map k a -> Map k a -> Map k a
1368 = Bin (size l + size r + 1) k x l r
1371 {--------------------------------------------------------------------
1372 Eq converts the tree to a list. In a lazy setting, this
1373 actually seems one of the faster methods to compare two trees
1374 and it is certainly the simplest :-)
1375 --------------------------------------------------------------------}
1376 instance (Eq k,Eq a) => Eq (Map k a) where
1377 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
1379 {--------------------------------------------------------------------
1381 --------------------------------------------------------------------}
1383 instance (Ord k, Ord v) => Ord (Map k v) where
1384 compare m1 m2 = compare (toAscList m1) (toAscList m2)
1386 {--------------------------------------------------------------------
1388 --------------------------------------------------------------------}
1389 instance Functor (Map k) where
1392 instance Traversable (Map k) where
1393 traverse f Tip = pure Tip
1394 traverse f (Bin s k v l r)
1395 = flip (Bin s k) <$> traverse f l <*> f v <*> traverse f r
1397 instance Foldable (Map k) where
1398 foldMap _f Tip = mempty
1399 foldMap f (Bin _s _k v l r)
1400 = foldMap f l `mappend` f v `mappend` foldMap f r
1402 {--------------------------------------------------------------------
1404 --------------------------------------------------------------------}
1405 instance (Ord k, Read k, Read e) => Read (Map k e) where
1406 #ifdef __GLASGOW_HASKELL__
1407 readPrec = parens $ prec 10 $ do
1408 Ident "fromList" <- lexP
1410 return (fromList xs)
1412 readListPrec = readListPrecDefault
1414 readsPrec p = readParen (p > 10) $ \ r -> do
1415 ("fromList",s) <- lex r
1417 return (fromList xs,t)
1420 -- parses a pair of things with the syntax a:=b
1421 readPair :: (Read a, Read b) => ReadS (a,b)
1422 readPair s = do (a, ct1) <- reads s
1423 (":=", ct2) <- lex ct1
1424 (b, ct3) <- reads ct2
1427 {--------------------------------------------------------------------
1429 --------------------------------------------------------------------}
1430 instance (Show k, Show a) => Show (Map k a) where
1431 showsPrec d m = showParen (d > 10) $
1432 showString "fromList " . shows (toList m)
1434 showMap :: (Show k,Show a) => [(k,a)] -> ShowS
1438 = showChar '{' . showElem x . showTail xs
1440 showTail [] = showChar '}'
1441 showTail (x:xs) = showString ", " . showElem x . showTail xs
1443 showElem (k,x) = shows k . showString " := " . shows x
1446 -- | /O(n)/. Show the tree that implements the map. The tree is shown
1447 -- in a compressed, hanging format.
1448 showTree :: (Show k,Show a) => Map k a -> String
1450 = showTreeWith showElem True False m
1452 showElem k x = show k ++ ":=" ++ show x
1455 {- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows
1456 the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
1457 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
1458 @wide@ is 'True', an extra wide version is shown.
1460 > Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
1461 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t
1468 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t
1479 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t
1491 showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
1492 showTreeWith showelem hang wide t
1493 | hang = (showsTreeHang showelem wide [] t) ""
1494 | otherwise = (showsTree showelem wide [] [] t) ""
1496 showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
1497 showsTree showelem wide lbars rbars t
1499 Tip -> showsBars lbars . showString "|\n"
1501 -> showsBars lbars . showString (showelem kx x) . showString "\n"
1503 -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
1504 showWide wide rbars .
1505 showsBars lbars . showString (showelem kx x) . showString "\n" .
1506 showWide wide lbars .
1507 showsTree showelem wide (withEmpty lbars) (withBar lbars) l
1509 showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
1510 showsTreeHang showelem wide bars t
1512 Tip -> showsBars bars . showString "|\n"
1514 -> showsBars bars . showString (showelem kx x) . showString "\n"
1516 -> showsBars bars . showString (showelem kx x) . showString "\n" .
1517 showWide wide bars .
1518 showsTreeHang showelem wide (withBar bars) l .
1519 showWide wide bars .
1520 showsTreeHang showelem wide (withEmpty bars) r
1524 | wide = showString (concat (reverse bars)) . showString "|\n"
1527 showsBars :: [String] -> ShowS
1531 _ -> showString (concat (reverse (tail bars))) . showString node
1534 withBar bars = "| ":bars
1535 withEmpty bars = " ":bars
1537 {--------------------------------------------------------------------
1539 --------------------------------------------------------------------}
1541 #include "Typeable.h"
1542 INSTANCE_TYPEABLE2(Map,mapTc,"Map")
1544 {--------------------------------------------------------------------
1546 --------------------------------------------------------------------}
1547 -- | /O(n)/. Test if the internal map structure is valid.
1548 valid :: Ord k => Map k a -> Bool
1550 = balanced t && ordered t && validsize t
1553 = bounded (const True) (const True) t
1558 Bin sz kx x l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r
1560 -- | Exported only for "Debug.QuickCheck"
1561 balanced :: Map k a -> Bool
1565 Bin sz kx x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
1566 balanced l && balanced r
1570 = (realsize t == Just (size t))
1575 Bin sz kx x l r -> case (realsize l,realsize r) of
1576 (Just n,Just m) | n+m+1 == sz -> Just sz
1579 {--------------------------------------------------------------------
1581 --------------------------------------------------------------------}
1585 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
1589 {--------------------------------------------------------------------
1591 --------------------------------------------------------------------}
1592 testTree xs = fromList [(x,"*") | x <- xs]
1593 test1 = testTree [1..20]
1594 test2 = testTree [30,29..10]
1595 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
1597 {--------------------------------------------------------------------
1599 --------------------------------------------------------------------}
1604 { configMaxTest = 500
1605 , configMaxFail = 5000
1606 , configSize = \n -> (div n 2 + 3)
1607 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
1611 {--------------------------------------------------------------------
1612 Arbitrary, reasonably balanced trees
1613 --------------------------------------------------------------------}
1614 instance (Enum k,Arbitrary a) => Arbitrary (Map k a) where
1615 arbitrary = sized (arbtree 0 maxkey)
1616 where maxkey = 10000
1618 arbtree :: (Enum k,Arbitrary a) => Int -> Int -> Int -> Gen (Map k a)
1620 | n <= 0 = return Tip
1621 | lo >= hi = return Tip
1622 | otherwise = do{ x <- arbitrary
1623 ; i <- choose (lo,hi)
1624 ; m <- choose (1,30)
1625 ; let (ml,mr) | m==(1::Int)= (1,2)
1629 ; l <- arbtree lo (i-1) (n `div` ml)
1630 ; r <- arbtree (i+1) hi (n `div` mr)
1631 ; return (bin (toEnum i) x l r)
1635 {--------------------------------------------------------------------
1637 --------------------------------------------------------------------}
1638 forValid :: (Show k,Enum k,Show a,Arbitrary a,Testable b) => (Map k a -> b) -> Property
1640 = forAll arbitrary $ \t ->
1641 -- classify (balanced t) "balanced" $
1642 classify (size t == 0) "empty" $
1643 classify (size t > 0 && size t <= 10) "small" $
1644 classify (size t > 10 && size t <= 64) "medium" $
1645 classify (size t > 64) "large" $
1648 forValidIntTree :: Testable a => (Map Int Int -> a) -> Property
1652 forValidUnitTree :: Testable a => (Map Int () -> a) -> Property
1658 = forValidUnitTree $ \t -> valid t
1660 {--------------------------------------------------------------------
1661 Single, Insert, Delete
1662 --------------------------------------------------------------------}
1663 prop_Single :: Int -> Int -> Bool
1665 = (insert k x empty == singleton k x)
1667 prop_InsertValid :: Int -> Property
1669 = forValidUnitTree $ \t -> valid (insert k () t)
1671 prop_InsertDelete :: Int -> Map Int () -> Property
1672 prop_InsertDelete k t
1673 = (lookup k t == Nothing) ==> delete k (insert k () t) == t
1675 prop_DeleteValid :: Int -> Property
1677 = forValidUnitTree $ \t ->
1678 valid (delete k (insert k () t))
1680 {--------------------------------------------------------------------
1682 --------------------------------------------------------------------}
1683 prop_Join :: Int -> Property
1685 = forValidUnitTree $ \t ->
1686 let (l,r) = split k t
1687 in valid (join k () l r)
1689 prop_Merge :: Int -> Property
1691 = forValidUnitTree $ \t ->
1692 let (l,r) = split k t
1693 in valid (merge l r)
1696 {--------------------------------------------------------------------
1698 --------------------------------------------------------------------}
1699 prop_UnionValid :: Property
1701 = forValidUnitTree $ \t1 ->
1702 forValidUnitTree $ \t2 ->
1705 prop_UnionInsert :: Int -> Int -> Map Int Int -> Bool
1706 prop_UnionInsert k x t
1707 = union (singleton k x) t == insert k x t
1709 prop_UnionAssoc :: Map Int Int -> Map Int Int -> Map Int Int -> Bool
1710 prop_UnionAssoc t1 t2 t3
1711 = union t1 (union t2 t3) == union (union t1 t2) t3
1713 prop_UnionComm :: Map Int Int -> Map Int Int -> Bool
1714 prop_UnionComm t1 t2
1715 = (union t1 t2 == unionWith (\x y -> y) t2 t1)
1718 = forValidIntTree $ \t1 ->
1719 forValidIntTree $ \t2 ->
1720 valid (unionWithKey (\k x y -> x+y) t1 t2)
1722 prop_UnionWith :: [(Int,Int)] -> [(Int,Int)] -> Bool
1723 prop_UnionWith xs ys
1724 = sum (elems (unionWith (+) (fromListWith (+) xs) (fromListWith (+) ys)))
1725 == (sum (Prelude.map snd xs) + sum (Prelude.map snd ys))
1728 = forValidUnitTree $ \t1 ->
1729 forValidUnitTree $ \t2 ->
1730 valid (difference t1 t2)
1732 prop_Diff :: [(Int,Int)] -> [(Int,Int)] -> Bool
1734 = List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
1735 == List.sort ((List.\\) (nub (Prelude.map fst xs)) (nub (Prelude.map fst ys)))
1738 = forValidUnitTree $ \t1 ->
1739 forValidUnitTree $ \t2 ->
1740 valid (intersection t1 t2)
1742 prop_Int :: [(Int,Int)] -> [(Int,Int)] -> Bool
1744 = List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
1745 == List.sort (nub ((List.intersect) (Prelude.map fst xs) (Prelude.map fst ys)))
1747 {--------------------------------------------------------------------
1749 --------------------------------------------------------------------}
1751 = forAll (choose (5,100)) $ \n ->
1752 let xs = [(x,()) | x <- [0..n::Int]]
1753 in fromAscList xs == fromList xs
1755 prop_List :: [Int] -> Bool
1757 = (sort (nub xs) == [x | (x,()) <- toList (fromList [(x,()) | x <- xs])])