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 -- Since many function names (but not the type name) clash with
15 -- "Prelude" names, this module is usually imported @qualified@, e.g.
17 -- > import Data.Map (Map)
18 -- > import qualified Data.Map as Map
20 -- The implementation of 'Map' is based on /size balanced/ binary trees (or
21 -- trees of /bounded balance/) as described by:
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>.
27 -- * J. Nievergelt and E.M. Reingold,
28 -- \"/Binary search trees of bounded balance/\",
29 -- SIAM journal of computing 2(1), March 1973.
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 -----------------------------------------------------------------------------
38 Map -- instance Eq,Show,Read
58 , insertWith, insertWithKey, insertLookupWithKey
119 , fromDistinctAscList
136 , isSubmapOf, isSubmapOfBy
137 , isProperSubmapOf, isProperSubmapOfBy
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(..))
171 import Control.Applicative (Applicative(..), (<$>))
172 import Data.Traversable (Traversable(traverse))
173 import Data.Foldable (Foldable(foldMap))
177 import qualified Prelude
178 import qualified List
179 import Debug.QuickCheck
180 import List(nub,sort)
183 #if __GLASGOW_HASKELL__
185 import Data.Generics.Basics
186 import Data.Generics.Instances
189 {--------------------------------------------------------------------
191 --------------------------------------------------------------------}
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
199 -- | /O(n+m)/. See 'difference'.
200 (\\) :: Ord k => Map k a -> Map k b -> Map k a
201 m1 \\ m2 = difference m1 m2
203 {--------------------------------------------------------------------
205 --------------------------------------------------------------------}
206 -- | A Map from keys @k@ to values @a@.
208 | Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
212 instance (Ord k) => Monoid (Map k v) where
217 #if __GLASGOW_HASKELL__
219 {--------------------------------------------------------------------
221 --------------------------------------------------------------------}
223 -- This instance preserves data abstraction at the cost of inefficiency.
224 -- We omit reflection services for the sake of data abstraction.
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
235 {--------------------------------------------------------------------
237 --------------------------------------------------------------------}
238 -- | /O(1)/. Is the map empty?
239 null :: Map k a -> Bool
243 Bin sz k x l r -> False
245 -- | /O(1)/. The number of elements in the map.
246 size :: Map k a -> Int
253 -- | /O(log n)/. Lookup the value at a key in the map.
256 -- @return@ the result in the monad or @fail@ in it the key isn't in the
257 -- map. Often, the monad to use is 'Maybe', so you get either
258 -- @('Just' result)@ or @'Nothing'@.
259 lookup :: (Monad m,Ord k) => k -> Map k a -> m a
260 lookup k t = case lookup' k t of
262 Nothing -> fail "Data.Map.lookup: Key not found"
263 lookup' :: Ord k => k -> Map k a -> Maybe a
268 -> case compare k kx of
273 lookupAssoc :: Ord k => k -> Map k a -> Maybe (k,a)
278 -> case compare k kx of
279 LT -> lookupAssoc k l
280 GT -> lookupAssoc k r
283 -- | /O(log n)/. Is the key a member of the map?
284 member :: Ord k => k -> Map k a -> Bool
290 -- | /O(log n)/. Is the key not a member of the map?
291 notMember :: Ord k => k -> Map k a -> Bool
292 notMember k m = not $ member k m
294 -- | /O(log n)/. Find the value at a key.
295 -- Calls 'error' when the element can not be found.
296 find :: Ord k => k -> Map k a -> a
299 Nothing -> error "Map.find: element not in the map"
302 -- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns
303 -- the value at key @k@ or returns @def@ when the key is not in the map.
304 findWithDefault :: Ord k => a -> k -> Map k a -> a
305 findWithDefault def k m
312 {--------------------------------------------------------------------
314 --------------------------------------------------------------------}
315 -- | /O(1)/. The empty map.
320 -- | /O(1)/. A map with a single element.
321 singleton :: k -> a -> Map k a
325 {--------------------------------------------------------------------
327 --------------------------------------------------------------------}
328 -- | /O(log n)/. Insert a new key and value in the map.
329 -- If the key is already present in the map, the associated value is
330 -- replaced with the supplied value, i.e. 'insert' is equivalent to
331 -- @'insertWith' 'const'@.
332 insert :: Ord k => k -> a -> Map k a -> Map k a
335 Tip -> singleton kx x
337 -> case compare kx ky of
338 LT -> balance ky y (insert kx x l) r
339 GT -> balance ky y l (insert kx x r)
340 EQ -> Bin sz kx x l r
342 -- | /O(log n)/. Insert with a combining function.
343 -- @'insertWith' f key value mp@
344 -- will insert the pair (key, value) into @mp@ if key does
345 -- not exist in the map. If the key does exist, the function will
346 -- insert the pair @(key, f new_value old_value)@.
347 insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
349 = insertWithKey (\k x y -> f x y) k x m
351 -- | /O(log n)/. Insert with a combining function.
352 -- @'insertWithKey' f key value mp@
353 -- will insert the pair (key, value) into @mp@ if key does
354 -- not exist in the map. If the key does exist, the function will
355 -- insert the pair @(key,f key new_value old_value)@.
356 -- Note that the key passed to f is the same key passed to 'insertWithKey'.
357 insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
358 insertWithKey f kx x t
360 Tip -> singleton kx x
362 -> case compare kx ky of
363 LT -> balance ky y (insertWithKey f kx x l) r
364 GT -> balance ky y l (insertWithKey f kx x r)
365 EQ -> Bin sy kx (f kx x y) l r
367 -- | /O(log n)/. The expression (@'insertLookupWithKey' f k x map@)
368 -- is a pair where the first element is equal to (@'lookup' k map@)
369 -- and the second element equal to (@'insertWithKey' f k x map@).
370 insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a,Map k a)
371 insertLookupWithKey f kx x t
373 Tip -> (Nothing, singleton kx x)
375 -> case compare kx ky of
376 LT -> let (found,l') = insertLookupWithKey f kx x l in (found,balance ky y l' r)
377 GT -> let (found,r') = insertLookupWithKey f kx x r in (found,balance ky y l r')
378 EQ -> (Just y, Bin sy kx (f kx x y) l r)
380 {--------------------------------------------------------------------
382 [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
383 --------------------------------------------------------------------}
384 -- | /O(log n)/. Delete a key and its value from the map. When the key is not
385 -- a member of the map, the original map is returned.
386 delete :: Ord k => k -> Map k a -> Map k a
391 -> case compare k kx of
392 LT -> balance kx x (delete k l) r
393 GT -> balance kx x l (delete k r)
396 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
397 -- a member of the map, the original map is returned.
398 adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
400 = adjustWithKey (\k x -> f x) k m
402 -- | /O(log n)/. Adjust a value at a specific key. When the key is not
403 -- a member of the map, the original map is returned.
404 adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
406 = updateWithKey (\k x -> Just (f k x)) k m
408 -- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@
409 -- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
410 -- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
411 update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
413 = updateWithKey (\k x -> f x) k m
415 -- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the
416 -- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',
417 -- the element is deleted. If it is (@'Just' y@), the key @k@ is bound
418 -- to the new value @y@.
419 updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
424 -> case compare k kx of
425 LT -> balance kx x (updateWithKey f k l) r
426 GT -> balance kx x l (updateWithKey f k r)
428 Just x' -> Bin sx kx x' l r
431 -- | /O(log n)/. Lookup and update.
432 updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
433 updateLookupWithKey f k t
437 -> case compare k kx of
438 LT -> let (found,l') = updateLookupWithKey f k l in (found,balance kx x l' r)
439 GT -> let (found,r') = updateLookupWithKey f k r in (found,balance kx x l r')
441 Just x' -> (Just x',Bin sx kx x' l r)
442 Nothing -> (Just x,glue l r)
444 -- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
445 -- 'alter' can be used to insert, delete, or update a value in a 'Map'.
446 -- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@
447 alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
450 Tip -> case f Nothing of
452 Just x -> singleton k x
454 -> case compare k kx of
455 LT -> balance kx x (alter f k l) r
456 GT -> balance kx x l (alter f k r)
457 EQ -> case f (Just x) of
458 Just x' -> Bin sx kx x' l r
461 {--------------------------------------------------------------------
463 --------------------------------------------------------------------}
464 -- | /O(log n)/. Return the /index/ of a key. The index is a number from
465 -- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
466 -- the key is not a 'member' of the map.
467 findIndex :: Ord k => k -> Map k a -> Int
469 = case lookupIndex k t of
470 Nothing -> error "Map.findIndex: element is not in the map"
473 -- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
474 -- /0/ up to, but not including, the 'size' of the map.
475 lookupIndex :: (Monad m,Ord k) => k -> Map k a -> m Int
476 lookupIndex k t = case lookup 0 t of
477 Nothing -> fail "Data.Map.lookupIndex: Key not found."
480 lookup idx Tip = Nothing
481 lookup idx (Bin _ kx x l r)
482 = case compare k kx of
484 GT -> lookup (idx + size l + 1) r
485 EQ -> Just (idx + size l)
487 -- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
488 -- invalid index is used.
489 elemAt :: Int -> Map k a -> (k,a)
490 elemAt i Tip = error "Map.elemAt: index out of range"
491 elemAt i (Bin _ kx x l r)
492 = case compare i sizeL of
494 GT -> elemAt (i-sizeL-1) r
499 -- | /O(log n)/. Update the element at /index/. Calls 'error' when an
500 -- invalid index is used.
501 updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
502 updateAt f i Tip = error "Map.updateAt: index out of range"
503 updateAt f i (Bin sx kx x l r)
504 = case compare i sizeL of
506 GT -> updateAt f (i-sizeL-1) r
508 Just x' -> Bin sx kx x' l r
513 -- | /O(log n)/. Delete the element at /index/.
514 -- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).
515 deleteAt :: Int -> Map k a -> Map k a
517 = updateAt (\k x -> Nothing) i map
520 {--------------------------------------------------------------------
522 --------------------------------------------------------------------}
523 -- | /O(log n)/. The minimal key of the map.
524 findMin :: Map k a -> (k,a)
525 findMin (Bin _ kx x Tip r) = (kx,x)
526 findMin (Bin _ kx x l r) = findMin l
527 findMin Tip = error "Map.findMin: empty map has no minimal element"
529 -- | /O(log n)/. The maximal key of the map.
530 findMax :: Map k a -> (k,a)
531 findMax (Bin _ kx x l Tip) = (kx,x)
532 findMax (Bin _ kx x l r) = findMax r
533 findMax Tip = error "Map.findMax: empty map has no maximal element"
535 -- | /O(log n)/. Delete the minimal key.
536 deleteMin :: Map k a -> Map k a
537 deleteMin (Bin _ kx x Tip r) = r
538 deleteMin (Bin _ kx x l r) = balance kx x (deleteMin l) r
541 -- | /O(log n)/. Delete the maximal key.
542 deleteMax :: Map k a -> Map k a
543 deleteMax (Bin _ kx x l Tip) = l
544 deleteMax (Bin _ kx x l r) = balance kx x l (deleteMax r)
547 -- | /O(log n)/. Update the value at the minimal key.
548 updateMin :: (a -> Maybe a) -> Map k a -> Map k a
550 = updateMinWithKey (\k x -> f x) m
552 -- | /O(log n)/. Update the value at the maximal key.
553 updateMax :: (a -> Maybe a) -> Map k a -> Map k a
555 = updateMaxWithKey (\k x -> f x) m
558 -- | /O(log n)/. Update the value at the minimal key.
559 updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
562 Bin sx kx x Tip r -> case f kx x of
564 Just x' -> Bin sx kx x' Tip r
565 Bin sx kx x l r -> balance kx x (updateMinWithKey f l) r
568 -- | /O(log n)/. Update the value at the maximal key.
569 updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
572 Bin sx kx x l Tip -> case f kx x of
574 Just x' -> Bin sx kx x' l Tip
575 Bin sx kx x l r -> balance kx x l (updateMaxWithKey f r)
578 -- | /O(log n)/. Retrieves the minimal key of the map, and the map stripped from that element
579 -- @fail@s (in the monad) when passed an empty map.
580 minView :: Monad m => Map k a -> m (Map k a, (k,a))
581 minView Tip = fail "Map.minView: empty map"
582 minView x = return (swap $ deleteFindMin x)
584 -- | /O(log n)/. Retrieves the maximal key of the map, and the map stripped from that element
585 -- @fail@s (in the monad) when passed an empty map.
586 maxView :: Monad m => Map k a -> m (Map k a, (k,a))
587 maxView Tip = fail "Map.maxView: empty map"
588 maxView x = return (swap $ deleteFindMax x)
592 {--------------------------------------------------------------------
594 --------------------------------------------------------------------}
595 -- | The union of a list of maps:
596 -- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).
597 unions :: Ord k => [Map k a] -> Map k a
599 = foldlStrict union empty ts
601 -- | The union of a list of maps, with a combining operation:
602 -- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).
603 unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a
605 = foldlStrict (unionWith f) empty ts
608 -- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.
609 -- It prefers @t1@ when duplicate keys are encountered,
610 -- i.e. (@'union' == 'unionWith' 'const'@).
611 -- The implementation uses the efficient /hedge-union/ algorithm.
612 -- Hedge-union is more efficient on (bigset `union` smallset)
613 union :: Ord k => Map k a -> Map k a -> Map k a
616 union t1 t2 = hedgeUnionL (const LT) (const GT) t1 t2
618 -- left-biased hedge union
619 hedgeUnionL cmplo cmphi t1 Tip
621 hedgeUnionL cmplo cmphi Tip (Bin _ kx x l r)
622 = join kx x (filterGt cmplo l) (filterLt cmphi r)
623 hedgeUnionL cmplo cmphi (Bin _ kx x l r) t2
624 = join kx x (hedgeUnionL cmplo cmpkx l (trim cmplo cmpkx t2))
625 (hedgeUnionL cmpkx cmphi r (trim cmpkx cmphi t2))
627 cmpkx k = compare kx k
629 -- right-biased hedge union
630 hedgeUnionR cmplo cmphi t1 Tip
632 hedgeUnionR cmplo cmphi Tip (Bin _ kx x l r)
633 = join kx x (filterGt cmplo l) (filterLt cmphi r)
634 hedgeUnionR cmplo cmphi (Bin _ kx x l r) t2
635 = join kx newx (hedgeUnionR cmplo cmpkx l lt)
636 (hedgeUnionR cmpkx cmphi r gt)
638 cmpkx k = compare kx k
639 lt = trim cmplo cmpkx t2
640 (found,gt) = trimLookupLo kx cmphi t2
645 {--------------------------------------------------------------------
646 Union with a combining function
647 --------------------------------------------------------------------}
648 -- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
649 unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
651 = unionWithKey (\k x y -> f x y) m1 m2
654 -- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
655 -- Hedge-union is more efficient on (bigset `union` smallset).
656 unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
657 unionWithKey f Tip t2 = t2
658 unionWithKey f t1 Tip = t1
659 unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
661 hedgeUnionWithKey f cmplo cmphi t1 Tip
663 hedgeUnionWithKey f cmplo cmphi Tip (Bin _ kx x l r)
664 = join kx x (filterGt cmplo l) (filterLt cmphi r)
665 hedgeUnionWithKey f cmplo cmphi (Bin _ kx x l r) t2
666 = join kx newx (hedgeUnionWithKey f cmplo cmpkx l lt)
667 (hedgeUnionWithKey f cmpkx cmphi r gt)
669 cmpkx k = compare kx k
670 lt = trim cmplo cmpkx t2
671 (found,gt) = trimLookupLo kx cmphi t2
674 Just (_,y) -> f kx x y
676 {--------------------------------------------------------------------
678 --------------------------------------------------------------------}
679 -- | /O(n+m)/. Difference of two maps.
680 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
681 difference :: Ord k => Map k a -> Map k b -> Map k a
682 difference Tip t2 = Tip
683 difference t1 Tip = t1
684 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
686 hedgeDiff cmplo cmphi Tip t
688 hedgeDiff cmplo cmphi (Bin _ kx x l r) Tip
689 = join kx x (filterGt cmplo l) (filterLt cmphi r)
690 hedgeDiff cmplo cmphi t (Bin _ kx x l r)
691 = merge (hedgeDiff cmplo cmpkx (trim cmplo cmpkx t) l)
692 (hedgeDiff cmpkx cmphi (trim cmpkx cmphi t) r)
694 cmpkx k = compare kx k
696 -- | /O(n+m)/. Difference with a combining function.
697 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
698 differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
699 differenceWith f m1 m2
700 = differenceWithKey (\k x y -> f x y) m1 m2
702 -- | /O(n+m)/. Difference with a combining function. When two equal keys are
703 -- encountered, the combining function is applied to the key and both values.
704 -- If it returns 'Nothing', the element is discarded (proper set difference). If
705 -- it returns (@'Just' y@), the element is updated with a new value @y@.
706 -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
707 differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
708 differenceWithKey f Tip t2 = Tip
709 differenceWithKey f t1 Tip = t1
710 differenceWithKey f t1 t2 = hedgeDiffWithKey f (const LT) (const GT) t1 t2
712 hedgeDiffWithKey f cmplo cmphi Tip t
714 hedgeDiffWithKey f cmplo cmphi (Bin _ kx x l r) Tip
715 = join kx x (filterGt cmplo l) (filterLt cmphi r)
716 hedgeDiffWithKey f cmplo cmphi t (Bin _ kx x l r)
718 Nothing -> merge tl tr
721 Nothing -> merge tl tr
722 Just z -> join ky z tl tr
724 cmpkx k = compare kx k
725 lt = trim cmplo cmpkx t
726 (found,gt) = trimLookupLo kx cmphi t
727 tl = hedgeDiffWithKey f cmplo cmpkx lt l
728 tr = hedgeDiffWithKey f cmpkx cmphi gt r
732 {--------------------------------------------------------------------
734 --------------------------------------------------------------------}
735 -- | /O(n+m)/. Intersection of two maps. The values in the first
736 -- map are returned, i.e. (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).
737 intersection :: Ord k => Map k a -> Map k b -> Map k a
739 = intersectionWithKey (\k x y -> x) m1 m2
741 -- | /O(n+m)/. Intersection with a combining function.
742 intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
743 intersectionWith f m1 m2
744 = intersectionWithKey (\k x y -> f x y) m1 m2
746 -- | /O(n+m)/. Intersection with a combining function.
747 -- Intersection is more efficient on (bigset `intersection` smallset)
748 --intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
749 --intersectionWithKey f Tip t = Tip
750 --intersectionWithKey f t Tip = Tip
751 --intersectionWithKey f t1 t2 = intersectWithKey f t1 t2
753 --intersectWithKey f Tip t = Tip
754 --intersectWithKey f t Tip = Tip
755 --intersectWithKey f t (Bin _ kx x l r)
757 -- Nothing -> merge tl tr
758 -- Just y -> join kx (f kx y x) tl tr
760 -- (lt,found,gt) = splitLookup kx t
761 -- tl = intersectWithKey f lt l
762 -- tr = intersectWithKey f gt r
765 intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
766 intersectionWithKey f Tip t = Tip
767 intersectionWithKey f t Tip = Tip
768 intersectionWithKey f t1@(Bin s1 k1 x1 l1 r1) t2@(Bin s2 k2 x2 l2 r2) =
770 let (lt,found,gt) = splitLookupWithKey k2 t1
771 tl = intersectionWithKey f lt l2
772 tr = intersectionWithKey f gt r2
774 Just (k,x) -> join k (f k x x2) tl tr
775 Nothing -> merge tl tr
776 else let (lt,found,gt) = splitLookup k1 t2
777 tl = intersectionWithKey f l1 lt
778 tr = intersectionWithKey f r1 gt
780 Just x -> join k1 (f k1 x1 x) tl tr
781 Nothing -> merge tl tr
785 {--------------------------------------------------------------------
787 --------------------------------------------------------------------}
789 -- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
790 isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
792 = isSubmapOfBy (==) m1 m2
795 The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if
796 all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when
797 applied to their respective values. For example, the following
798 expressions are all 'True':
800 > isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
801 > isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
802 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
804 But the following are all 'False':
806 > isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
807 > isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
808 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
810 isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool
812 = (size t1 <= size t2) && (submap' f t1 t2)
814 submap' f Tip t = True
815 submap' f t Tip = False
816 submap' f (Bin _ kx x l r) t
819 Just y -> f x y && submap' f l lt && submap' f r gt
821 (lt,found,gt) = splitLookup kx t
823 -- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
824 -- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
825 isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
826 isProperSubmapOf m1 m2
827 = isProperSubmapOfBy (==) m1 m2
829 {- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
830 The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
831 @m1@ and @m2@ are not equal,
832 all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
833 applied to their respective values. For example, the following
834 expressions are all 'True':
836 > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
837 > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
839 But the following are all 'False':
841 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
842 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
843 > isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
845 isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
846 isProperSubmapOfBy f t1 t2
847 = (size t1 < size t2) && (submap' f t1 t2)
849 {--------------------------------------------------------------------
851 --------------------------------------------------------------------}
852 -- | /O(n)/. Filter all values that satisfy the predicate.
853 filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
855 = filterWithKey (\k x -> p x) m
857 -- | /O(n)/. Filter all keys\/values that satisfy the predicate.
858 filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
859 filterWithKey p Tip = Tip
860 filterWithKey p (Bin _ kx x l r)
861 | p kx x = join kx x (filterWithKey p l) (filterWithKey p r)
862 | otherwise = merge (filterWithKey p l) (filterWithKey p r)
865 -- | /O(n)/. partition the map according to a predicate. The first
866 -- map contains all elements that satisfy the predicate, the second all
867 -- elements that fail the predicate. See also 'split'.
868 partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a)
870 = partitionWithKey (\k x -> p x) m
872 -- | /O(n)/. partition the map according to a predicate. The first
873 -- map contains all elements that satisfy the predicate, the second all
874 -- elements that fail the predicate. See also 'split'.
875 partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
876 partitionWithKey p Tip = (Tip,Tip)
877 partitionWithKey p (Bin _ kx x l r)
878 | p kx x = (join kx x l1 r1,merge l2 r2)
879 | otherwise = (merge l1 r1,join kx x l2 r2)
881 (l1,l2) = partitionWithKey p l
882 (r1,r2) = partitionWithKey p r
884 -- | /O(n)/. Map values and collect the 'Just' results.
885 mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b
887 = mapMaybeWithKey (\k x -> f x) m
889 -- | /O(n)/. Map keys\/values and collect the 'Just' results.
890 mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b
891 mapMaybeWithKey f Tip = Tip
892 mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of
893 Just y -> join kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
894 Nothing -> merge (mapMaybeWithKey f l) (mapMaybeWithKey f r)
896 -- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
897 mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)
899 = mapEitherWithKey (\k x -> f x) m
901 -- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
902 mapEitherWithKey :: Ord k =>
903 (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
904 mapEitherWithKey f Tip = (Tip, Tip)
905 mapEitherWithKey f (Bin _ kx x l r) = case f kx x of
906 Left y -> (join kx y l1 r1, merge l2 r2)
907 Right z -> (merge l1 r1, join kx z l2 r2)
909 (l1,l2) = mapEitherWithKey f l
910 (r1,r2) = mapEitherWithKey f r
912 {--------------------------------------------------------------------
914 --------------------------------------------------------------------}
915 -- | /O(n)/. Map a function over all values in the map.
916 map :: (a -> b) -> Map k a -> Map k b
918 = mapWithKey (\k x -> f x) m
920 -- | /O(n)/. Map a function over all values in the map.
921 mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
922 mapWithKey f Tip = Tip
923 mapWithKey f (Bin sx kx x l r)
924 = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)
926 -- | /O(n)/. The function 'mapAccum' threads an accumulating
927 -- argument through the map in ascending order of keys.
928 mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
930 = mapAccumWithKey (\a k x -> f a x) a m
932 -- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating
933 -- argument through the map in ascending order of keys.
934 mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
935 mapAccumWithKey f a t
938 -- | /O(n)/. The function 'mapAccumL' threads an accumulating
939 -- argument throught the map in ascending order of keys.
940 mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
945 -> let (a1,l') = mapAccumL f a l
947 (a3,r') = mapAccumL f a2 r
948 in (a3,Bin sx kx x' l' r')
950 -- | /O(n)/. The function 'mapAccumR' threads an accumulating
951 -- argument throught the map in descending order of keys.
952 mapAccumR :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
957 -> let (a1,r') = mapAccumR f a r
959 (a3,l') = mapAccumR f a2 l
960 in (a3,Bin sx kx x' l' r')
963 -- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
965 -- The size of the result may be smaller if @f@ maps two or more distinct
966 -- keys to the same new key. In this case the value at the smallest of
967 -- these keys is retained.
969 mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
970 mapKeys = mapKeysWith (\x y->x)
973 -- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
975 -- The size of the result may be smaller if @f@ maps two or more distinct
976 -- keys to the same new key. In this case the associated values will be
977 -- combined using @c@.
979 mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a
980 mapKeysWith c f = fromListWith c . List.map fFirst . toList
981 where fFirst (x,y) = (f x, y)
985 -- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
986 -- is strictly monotonic.
987 -- /The precondition is not checked./
988 -- Semi-formally, we have:
990 -- > and [x < y ==> f x < f y | x <- ls, y <- ls]
991 -- > ==> mapKeysMonotonic f s == mapKeys f s
992 -- > where ls = keys s
994 mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a
995 mapKeysMonotonic f Tip = Tip
996 mapKeysMonotonic f (Bin sz k x l r) =
997 Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)
999 {--------------------------------------------------------------------
1001 --------------------------------------------------------------------}
1003 -- | /O(n)/. Fold the values in the map, such that
1004 -- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.
1007 -- > elems map = fold (:) [] map
1009 fold :: (a -> b -> b) -> b -> Map k a -> b
1011 = foldWithKey (\k x z -> f x z) z m
1013 -- | /O(n)/. Fold the keys and values in the map, such that
1014 -- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
1017 -- > keys map = foldWithKey (\k x ks -> k:ks) [] map
1019 foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
1023 -- | /O(n)/. In-order fold.
1024 foldi :: (k -> a -> b -> b -> b) -> b -> Map k a -> b
1026 foldi f z (Bin _ kx x l r) = f kx x (foldi f z l) (foldi f z r)
1028 -- | /O(n)/. Post-order fold.
1029 foldr :: (k -> a -> b -> b) -> b -> Map k a -> b
1031 foldr f z (Bin _ kx x l r) = foldr f (f kx x (foldr f z r)) l
1033 -- | /O(n)/. Pre-order fold.
1034 foldl :: (b -> k -> a -> b) -> b -> Map k a -> b
1036 foldl f z (Bin _ kx x l r) = foldl f (f (foldl f z l) kx x) r
1038 {--------------------------------------------------------------------
1040 --------------------------------------------------------------------}
1042 -- Return all elements of the map in the ascending order of their keys.
1043 elems :: Map k a -> [a]
1045 = [x | (k,x) <- assocs m]
1047 -- | /O(n)/. Return all keys of the map in ascending order.
1048 keys :: Map k a -> [k]
1050 = [k | (k,x) <- assocs m]
1052 -- | /O(n)/. The set of all keys of the map.
1053 keysSet :: Map k a -> Set.Set k
1054 keysSet m = Set.fromDistinctAscList (keys m)
1056 -- | /O(n)/. Return all key\/value pairs in the map in ascending key order.
1057 assocs :: Map k a -> [(k,a)]
1061 {--------------------------------------------------------------------
1063 use [foldlStrict] to reduce demand on the control-stack
1064 --------------------------------------------------------------------}
1065 -- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
1066 fromList :: Ord k => [(k,a)] -> Map k a
1068 = foldlStrict ins empty xs
1070 ins t (k,x) = insert k x t
1072 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
1073 fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a
1075 = fromListWithKey (\k x y -> f x y) xs
1077 -- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
1078 fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1079 fromListWithKey f xs
1080 = foldlStrict ins empty xs
1082 ins t (k,x) = insertWithKey f k x t
1084 -- | /O(n)/. Convert to a list of key\/value pairs.
1085 toList :: Map k a -> [(k,a)]
1086 toList t = toAscList t
1088 -- | /O(n)/. Convert to an ascending list.
1089 toAscList :: Map k a -> [(k,a)]
1090 toAscList t = foldr (\k x xs -> (k,x):xs) [] t
1093 toDescList :: Map k a -> [(k,a)]
1094 toDescList t = foldl (\xs k x -> (k,x):xs) [] t
1097 {--------------------------------------------------------------------
1098 Building trees from ascending/descending lists can be done in linear time.
1100 Note that if [xs] is ascending that:
1101 fromAscList xs == fromList xs
1102 fromAscListWith f xs == fromListWith f xs
1103 --------------------------------------------------------------------}
1104 -- | /O(n)/. Build a map from an ascending list in linear time.
1105 -- /The precondition (input list is ascending) is not checked./
1106 fromAscList :: Eq k => [(k,a)] -> Map k a
1108 = fromAscListWithKey (\k x y -> x) xs
1110 -- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
1111 -- /The precondition (input list is ascending) is not checked./
1112 fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
1113 fromAscListWith f xs
1114 = fromAscListWithKey (\k x y -> f x y) xs
1116 -- | /O(n)/. Build a map from an ascending list in linear time with a
1117 -- combining function for equal keys.
1118 -- /The precondition (input list is ascending) is not checked./
1119 fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
1120 fromAscListWithKey f xs
1121 = fromDistinctAscList (combineEq f xs)
1123 -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
1128 (x:xx) -> combineEq' x xx
1130 combineEq' z [] = [z]
1131 combineEq' z@(kz,zz) (x@(kx,xx):xs)
1132 | kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs
1133 | otherwise = z:combineEq' x xs
1136 -- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
1137 -- /The precondition is not checked./
1138 fromDistinctAscList :: [(k,a)] -> Map k a
1139 fromDistinctAscList xs
1140 = build const (length xs) xs
1142 -- 1) use continutations so that we use heap space instead of stack space.
1143 -- 2) special case for n==5 to build bushier trees.
1144 build c 0 xs = c Tip xs
1145 build c 5 xs = case xs of
1146 ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
1147 -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
1148 build c n xs = seq nr $ build (buildR nr c) nl xs
1153 buildR n c l ((k,x):ys) = build (buildB l k x c) n ys
1154 buildB l k x c r zs = c (bin k x l r) zs
1158 {--------------------------------------------------------------------
1159 Utility functions that return sub-ranges of the original
1160 tree. Some functions take a comparison function as argument to
1161 allow comparisons against infinite values. A function [cmplo k]
1162 should be read as [compare lo k].
1164 [trim cmplo cmphi t] A tree that is either empty or where [cmplo k == LT]
1165 and [cmphi k == GT] for the key [k] of the root.
1166 [filterGt cmp t] A tree where for all keys [k]. [cmp k == LT]
1167 [filterLt cmp t] A tree where for all keys [k]. [cmp k == GT]
1169 [split k t] Returns two trees [l] and [r] where all keys
1170 in [l] are <[k] and all keys in [r] are >[k].
1171 [splitLookup k t] Just like [split] but also returns whether [k]
1172 was found in the tree.
1173 --------------------------------------------------------------------}
1175 {--------------------------------------------------------------------
1176 [trim lo hi t] trims away all subtrees that surely contain no
1177 values between the range [lo] to [hi]. The returned tree is either
1178 empty or the key of the root is between @lo@ and @hi@.
1179 --------------------------------------------------------------------}
1180 trim :: (k -> Ordering) -> (k -> Ordering) -> Map k a -> Map k a
1181 trim cmplo cmphi Tip = Tip
1182 trim cmplo cmphi t@(Bin sx kx x l r)
1184 LT -> case cmphi kx of
1186 le -> trim cmplo cmphi l
1187 ge -> trim cmplo cmphi r
1189 trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a)
1190 trimLookupLo lo cmphi Tip = (Nothing,Tip)
1191 trimLookupLo lo cmphi t@(Bin sx kx x l r)
1192 = case compare lo kx of
1193 LT -> case cmphi kx of
1194 GT -> (lookupAssoc lo t, t)
1195 le -> trimLookupLo lo cmphi l
1196 GT -> trimLookupLo lo cmphi r
1197 EQ -> (Just (kx,x),trim (compare lo) cmphi r)
1200 {--------------------------------------------------------------------
1201 [filterGt k t] filter all keys >[k] from tree [t]
1202 [filterLt k t] filter all keys <[k] from tree [t]
1203 --------------------------------------------------------------------}
1204 filterGt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1205 filterGt cmp Tip = Tip
1206 filterGt cmp (Bin sx kx x l r)
1208 LT -> join kx x (filterGt cmp l) r
1209 GT -> filterGt cmp r
1212 filterLt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
1213 filterLt cmp Tip = Tip
1214 filterLt cmp (Bin sx kx x l r)
1216 LT -> filterLt cmp l
1217 GT -> join kx x l (filterLt cmp r)
1220 {--------------------------------------------------------------------
1222 --------------------------------------------------------------------}
1223 -- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where
1224 -- 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@.
1225 split :: Ord k => k -> Map k a -> (Map k a,Map k a)
1226 split k Tip = (Tip,Tip)
1227 split k (Bin sx kx x l r)
1228 = case compare k kx of
1229 LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
1230 GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
1233 -- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just
1234 -- like 'split' but also returns @'lookup' k map@.
1235 splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)
1236 splitLookup k Tip = (Tip,Nothing,Tip)
1237 splitLookup k (Bin sx kx x l r)
1238 = case compare k kx of
1239 LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
1240 GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
1244 splitLookupWithKey :: Ord k => k -> Map k a -> (Map k a,Maybe (k,a),Map k a)
1245 splitLookupWithKey k Tip = (Tip,Nothing,Tip)
1246 splitLookupWithKey k (Bin sx kx x l r)
1247 = case compare k kx of
1248 LT -> let (lt,z,gt) = splitLookupWithKey k l in (lt,z,join kx x gt r)
1249 GT -> let (lt,z,gt) = splitLookupWithKey k r in (join kx x l lt,z,gt)
1250 EQ -> (l,Just (kx, x),r)
1252 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
1253 -- element was found in the original set.
1254 splitMember :: Ord k => k -> Map k a -> (Map k a,Bool,Map k a)
1255 splitMember x t = let (l,m,r) = splitLookup x t in
1256 (l,maybe False (const True) m,r)
1259 {--------------------------------------------------------------------
1260 Utility functions that maintain the balance properties of the tree.
1261 All constructors assume that all values in [l] < [k] and all values
1262 in [r] > [k], and that [l] and [r] are valid trees.
1264 In order of sophistication:
1265 [Bin sz k x l r] The type constructor.
1266 [bin k x l r] Maintains the correct size, assumes that both [l]
1267 and [r] are balanced with respect to each other.
1268 [balance k x l r] Restores the balance and size.
1269 Assumes that the original tree was balanced and
1270 that [l] or [r] has changed by at most one element.
1271 [join k x l r] Restores balance and size.
1273 Furthermore, we can construct a new tree from two trees. Both operations
1274 assume that all values in [l] < all values in [r] and that [l] and [r]
1276 [glue l r] Glues [l] and [r] together. Assumes that [l] and
1277 [r] are already balanced with respect to each other.
1278 [merge l r] Merges two trees and restores balance.
1280 Note: in contrast to Adam's paper, we use (<=) comparisons instead
1281 of (<) comparisons in [join], [merge] and [balance].
1282 Quickcheck (on [difference]) showed that this was necessary in order
1283 to maintain the invariants. It is quite unsatisfactory that I haven't
1284 been able to find out why this is actually the case! Fortunately, it
1285 doesn't hurt to be a bit more conservative.
1286 --------------------------------------------------------------------}
1288 {--------------------------------------------------------------------
1290 --------------------------------------------------------------------}
1291 join :: Ord k => k -> a -> Map k a -> Map k a -> Map k a
1292 join kx x Tip r = insertMin kx x r
1293 join kx x l Tip = insertMax kx x l
1294 join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
1295 | delta*sizeL <= sizeR = balance kz z (join kx x l lz) rz
1296 | delta*sizeR <= sizeL = balance ky y ly (join kx x ry r)
1297 | otherwise = bin kx x l r
1300 -- insertMin and insertMax don't perform potentially expensive comparisons.
1301 insertMax,insertMin :: k -> a -> Map k a -> Map k a
1304 Tip -> singleton kx x
1306 -> balance ky y l (insertMax kx x r)
1310 Tip -> singleton kx x
1312 -> balance ky y (insertMin kx x l) r
1314 {--------------------------------------------------------------------
1315 [merge l r]: merges two trees.
1316 --------------------------------------------------------------------}
1317 merge :: Map k a -> Map k a -> Map k a
1320 merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
1321 | delta*sizeL <= sizeR = balance ky y (merge l ly) ry
1322 | delta*sizeR <= sizeL = balance kx x lx (merge rx r)
1323 | otherwise = glue l r
1325 {--------------------------------------------------------------------
1326 [glue l r]: glues two trees together.
1327 Assumes that [l] and [r] are already balanced with respect to each other.
1328 --------------------------------------------------------------------}
1329 glue :: Map k a -> Map k a -> Map k a
1333 | size l > size r = let ((km,m),l') = deleteFindMax l in balance km m l' r
1334 | otherwise = let ((km,m),r') = deleteFindMin r in balance km m l r'
1337 -- | /O(log n)/. Delete and find the minimal element.
1338 deleteFindMin :: Map k a -> ((k,a),Map k a)
1341 Bin _ k x Tip r -> ((k,x),r)
1342 Bin _ k x l r -> let (km,l') = deleteFindMin l in (km,balance k x l' r)
1343 Tip -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)
1345 -- | /O(log n)/. Delete and find the maximal element.
1346 deleteFindMax :: Map k a -> ((k,a),Map k a)
1349 Bin _ k x l Tip -> ((k,x),l)
1350 Bin _ k x l r -> let (km,r') = deleteFindMax r in (km,balance k x l r')
1351 Tip -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
1354 {--------------------------------------------------------------------
1355 [balance l x r] balances two trees with value x.
1356 The sizes of the trees should balance after decreasing the
1357 size of one of them. (a rotation).
1359 [delta] is the maximal relative difference between the sizes of
1360 two trees, it corresponds with the [w] in Adams' paper.
1361 [ratio] is the ratio between an outer and inner sibling of the
1362 heavier subtree in an unbalanced setting. It determines
1363 whether a double or single rotation should be performed
1364 to restore balance. It is correspondes with the inverse
1365 of $\alpha$ in Adam's article.
1368 - [delta] should be larger than 4.646 with a [ratio] of 2.
1369 - [delta] should be larger than 3.745 with a [ratio] of 1.534.
1371 - A lower [delta] leads to a more 'perfectly' balanced tree.
1372 - A higher [delta] performs less rebalancing.
1374 - Balancing is automatic for random data and a balancing
1375 scheme is only necessary to avoid pathological worst cases.
1376 Almost any choice will do, and in practice, a rather large
1377 [delta] may perform better than smaller one.
1379 Note: in contrast to Adam's paper, we use a ratio of (at least) [2]
1380 to decide whether a single or double rotation is needed. Allthough
1381 he actually proves that this ratio is needed to maintain the
1382 invariants, his implementation uses an invalid ratio of [1].
1383 --------------------------------------------------------------------}
1388 balance :: k -> a -> Map k a -> Map k a -> Map k a
1390 | sizeL + sizeR <= 1 = Bin sizeX k x l r
1391 | sizeR >= delta*sizeL = rotateL k x l r
1392 | sizeL >= delta*sizeR = rotateR k x l r
1393 | otherwise = Bin sizeX k x l r
1397 sizeX = sizeL + sizeR + 1
1400 rotateL k x l r@(Bin _ _ _ ly ry)
1401 | size ly < ratio*size ry = singleL k x l r
1402 | otherwise = doubleL k x l r
1404 rotateR k x l@(Bin _ _ _ ly ry) r
1405 | size ry < ratio*size ly = singleR k x l r
1406 | otherwise = doubleR k x l r
1409 singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
1410 singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)
1412 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)
1413 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)
1416 {--------------------------------------------------------------------
1417 The bin constructor maintains the size of the tree
1418 --------------------------------------------------------------------}
1419 bin :: k -> a -> Map k a -> Map k a -> Map k a
1421 = Bin (size l + size r + 1) k x l r
1424 {--------------------------------------------------------------------
1425 Eq converts the tree to a list. In a lazy setting, this
1426 actually seems one of the faster methods to compare two trees
1427 and it is certainly the simplest :-)
1428 --------------------------------------------------------------------}
1429 instance (Eq k,Eq a) => Eq (Map k a) where
1430 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
1432 {--------------------------------------------------------------------
1434 --------------------------------------------------------------------}
1436 instance (Ord k, Ord v) => Ord (Map k v) where
1437 compare m1 m2 = compare (toAscList m1) (toAscList m2)
1439 {--------------------------------------------------------------------
1441 --------------------------------------------------------------------}
1442 instance Functor (Map k) where
1445 instance Traversable (Map k) where
1446 traverse f Tip = pure Tip
1447 traverse f (Bin s k v l r)
1448 = flip (Bin s k) <$> traverse f l <*> f v <*> traverse f r
1450 instance Foldable (Map k) where
1451 foldMap _f Tip = mempty
1452 foldMap f (Bin _s _k v l r)
1453 = foldMap f l `mappend` f v `mappend` foldMap f r
1455 {--------------------------------------------------------------------
1457 --------------------------------------------------------------------}
1458 instance (Ord k, Read k, Read e) => Read (Map k e) where
1459 #ifdef __GLASGOW_HASKELL__
1460 readPrec = parens $ prec 10 $ do
1461 Ident "fromList" <- lexP
1463 return (fromList xs)
1465 readListPrec = readListPrecDefault
1467 readsPrec p = readParen (p > 10) $ \ r -> do
1468 ("fromList",s) <- lex r
1470 return (fromList xs,t)
1473 -- parses a pair of things with the syntax a:=b
1474 readPair :: (Read a, Read b) => ReadS (a,b)
1475 readPair s = do (a, ct1) <- reads s
1476 (":=", ct2) <- lex ct1
1477 (b, ct3) <- reads ct2
1480 {--------------------------------------------------------------------
1482 --------------------------------------------------------------------}
1483 instance (Show k, Show a) => Show (Map k a) where
1484 showsPrec d m = showParen (d > 10) $
1485 showString "fromList " . shows (toList m)
1487 showMap :: (Show k,Show a) => [(k,a)] -> ShowS
1491 = showChar '{' . showElem x . showTail xs
1493 showTail [] = showChar '}'
1494 showTail (x:xs) = showString ", " . showElem x . showTail xs
1496 showElem (k,x) = shows k . showString " := " . shows x
1499 -- | /O(n)/. Show the tree that implements the map. The tree is shown
1500 -- in a compressed, hanging format.
1501 showTree :: (Show k,Show a) => Map k a -> String
1503 = showTreeWith showElem True False m
1505 showElem k x = show k ++ ":=" ++ show x
1508 {- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows
1509 the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
1510 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
1511 @wide@ is 'True', an extra wide version is shown.
1513 > Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
1514 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t
1521 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t
1532 > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t
1544 showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
1545 showTreeWith showelem hang wide t
1546 | hang = (showsTreeHang showelem wide [] t) ""
1547 | otherwise = (showsTree showelem wide [] [] t) ""
1549 showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
1550 showsTree showelem wide lbars rbars t
1552 Tip -> showsBars lbars . showString "|\n"
1554 -> showsBars lbars . showString (showelem kx x) . showString "\n"
1556 -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
1557 showWide wide rbars .
1558 showsBars lbars . showString (showelem kx x) . showString "\n" .
1559 showWide wide lbars .
1560 showsTree showelem wide (withEmpty lbars) (withBar lbars) l
1562 showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
1563 showsTreeHang showelem wide bars t
1565 Tip -> showsBars bars . showString "|\n"
1567 -> showsBars bars . showString (showelem kx x) . showString "\n"
1569 -> showsBars bars . showString (showelem kx x) . showString "\n" .
1570 showWide wide bars .
1571 showsTreeHang showelem wide (withBar bars) l .
1572 showWide wide bars .
1573 showsTreeHang showelem wide (withEmpty bars) r
1577 | wide = showString (concat (reverse bars)) . showString "|\n"
1580 showsBars :: [String] -> ShowS
1584 _ -> showString (concat (reverse (tail bars))) . showString node
1587 withBar bars = "| ":bars
1588 withEmpty bars = " ":bars
1590 {--------------------------------------------------------------------
1592 --------------------------------------------------------------------}
1594 #include "Typeable.h"
1595 INSTANCE_TYPEABLE2(Map,mapTc,"Map")
1597 {--------------------------------------------------------------------
1599 --------------------------------------------------------------------}
1600 -- | /O(n)/. Test if the internal map structure is valid.
1601 valid :: Ord k => Map k a -> Bool
1603 = balanced t && ordered t && validsize t
1606 = bounded (const True) (const True) t
1611 Bin sz kx x l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r
1613 -- | Exported only for "Debug.QuickCheck"
1614 balanced :: Map k a -> Bool
1618 Bin sz kx x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
1619 balanced l && balanced r
1623 = (realsize t == Just (size t))
1628 Bin sz kx x l r -> case (realsize l,realsize r) of
1629 (Just n,Just m) | n+m+1 == sz -> Just sz
1632 {--------------------------------------------------------------------
1634 --------------------------------------------------------------------}
1638 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
1642 {--------------------------------------------------------------------
1644 --------------------------------------------------------------------}
1645 testTree xs = fromList [(x,"*") | x <- xs]
1646 test1 = testTree [1..20]
1647 test2 = testTree [30,29..10]
1648 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
1650 {--------------------------------------------------------------------
1652 --------------------------------------------------------------------}
1657 { configMaxTest = 500
1658 , configMaxFail = 5000
1659 , configSize = \n -> (div n 2 + 3)
1660 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
1664 {--------------------------------------------------------------------
1665 Arbitrary, reasonably balanced trees
1666 --------------------------------------------------------------------}
1667 instance (Enum k,Arbitrary a) => Arbitrary (Map k a) where
1668 arbitrary = sized (arbtree 0 maxkey)
1669 where maxkey = 10000
1671 arbtree :: (Enum k,Arbitrary a) => Int -> Int -> Int -> Gen (Map k a)
1673 | n <= 0 = return Tip
1674 | lo >= hi = return Tip
1675 | otherwise = do{ x <- arbitrary
1676 ; i <- choose (lo,hi)
1677 ; m <- choose (1,30)
1678 ; let (ml,mr) | m==(1::Int)= (1,2)
1682 ; l <- arbtree lo (i-1) (n `div` ml)
1683 ; r <- arbtree (i+1) hi (n `div` mr)
1684 ; return (bin (toEnum i) x l r)
1688 {--------------------------------------------------------------------
1690 --------------------------------------------------------------------}
1691 forValid :: (Show k,Enum k,Show a,Arbitrary a,Testable b) => (Map k a -> b) -> Property
1693 = forAll arbitrary $ \t ->
1694 -- classify (balanced t) "balanced" $
1695 classify (size t == 0) "empty" $
1696 classify (size t > 0 && size t <= 10) "small" $
1697 classify (size t > 10 && size t <= 64) "medium" $
1698 classify (size t > 64) "large" $
1701 forValidIntTree :: Testable a => (Map Int Int -> a) -> Property
1705 forValidUnitTree :: Testable a => (Map Int () -> a) -> Property
1711 = forValidUnitTree $ \t -> valid t
1713 {--------------------------------------------------------------------
1714 Single, Insert, Delete
1715 --------------------------------------------------------------------}
1716 prop_Single :: Int -> Int -> Bool
1718 = (insert k x empty == singleton k x)
1720 prop_InsertValid :: Int -> Property
1722 = forValidUnitTree $ \t -> valid (insert k () t)
1724 prop_InsertDelete :: Int -> Map Int () -> Property
1725 prop_InsertDelete k t
1726 = (lookup k t == Nothing) ==> delete k (insert k () t) == t
1728 prop_DeleteValid :: Int -> Property
1730 = forValidUnitTree $ \t ->
1731 valid (delete k (insert k () t))
1733 {--------------------------------------------------------------------
1735 --------------------------------------------------------------------}
1736 prop_Join :: Int -> Property
1738 = forValidUnitTree $ \t ->
1739 let (l,r) = split k t
1740 in valid (join k () l r)
1742 prop_Merge :: Int -> Property
1744 = forValidUnitTree $ \t ->
1745 let (l,r) = split k t
1746 in valid (merge l r)
1749 {--------------------------------------------------------------------
1751 --------------------------------------------------------------------}
1752 prop_UnionValid :: Property
1754 = forValidUnitTree $ \t1 ->
1755 forValidUnitTree $ \t2 ->
1758 prop_UnionInsert :: Int -> Int -> Map Int Int -> Bool
1759 prop_UnionInsert k x t
1760 = union (singleton k x) t == insert k x t
1762 prop_UnionAssoc :: Map Int Int -> Map Int Int -> Map Int Int -> Bool
1763 prop_UnionAssoc t1 t2 t3
1764 = union t1 (union t2 t3) == union (union t1 t2) t3
1766 prop_UnionComm :: Map Int Int -> Map Int Int -> Bool
1767 prop_UnionComm t1 t2
1768 = (union t1 t2 == unionWith (\x y -> y) t2 t1)
1771 = forValidIntTree $ \t1 ->
1772 forValidIntTree $ \t2 ->
1773 valid (unionWithKey (\k x y -> x+y) t1 t2)
1775 prop_UnionWith :: [(Int,Int)] -> [(Int,Int)] -> Bool
1776 prop_UnionWith xs ys
1777 = sum (elems (unionWith (+) (fromListWith (+) xs) (fromListWith (+) ys)))
1778 == (sum (Prelude.map snd xs) + sum (Prelude.map snd ys))
1781 = forValidUnitTree $ \t1 ->
1782 forValidUnitTree $ \t2 ->
1783 valid (difference t1 t2)
1785 prop_Diff :: [(Int,Int)] -> [(Int,Int)] -> Bool
1787 = List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
1788 == List.sort ((List.\\) (nub (Prelude.map fst xs)) (nub (Prelude.map fst ys)))
1791 = forValidUnitTree $ \t1 ->
1792 forValidUnitTree $ \t2 ->
1793 valid (intersection t1 t2)
1795 prop_Int :: [(Int,Int)] -> [(Int,Int)] -> Bool
1797 = List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
1798 == List.sort (nub ((List.intersect) (Prelude.map fst xs) (Prelude.map fst ys)))
1800 {--------------------------------------------------------------------
1802 --------------------------------------------------------------------}
1804 = forAll (choose (5,100)) $ \n ->
1805 let xs = [(x,()) | x <- [0..n::Int]]
1806 in fromAscList xs == fromList xs
1808 prop_List :: [Int] -> Bool
1810 = (sort (nub xs) == [x | (x,()) <- toList (fromList [(x,()) | x <- xs])])