-- * J. Nievergelt and E.M. Reingold,
-- \"/Binary search trees of bounded balance/\",
-- SIAM journal of computing 2(1), March 1973.
+--
+-- Note that the implementation is /left-biased/ -- the elements of a
+-- first argument are always preferred to the second, for example in
+-- 'union' or 'insert'.
-----------------------------------------------------------------------------
module Data.Map (
GT -> lookup' k r
EQ -> Just x
+lookupAssoc :: Ord k => k -> Map k a -> Maybe (k,a)
+lookupAssoc k t
+ = case t of
+ Tip -> Nothing
+ Bin sz kx x l r
+ -> case compare k kx of
+ LT -> lookupAssoc k l
+ GT -> lookupAssoc k r
+ EQ -> Just (kx,x)
+
-- | /O(log n)/. Is the key a member of the map?
member :: Ord k => k -> Map k a -> Bool
member k m
-- @'insertWith' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
--- insert @f new_value old_value@.
+-- insert the pair @(key, f new_value old_value)@.
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWith f k x m
= insertWithKey (\k x y -> f x y) k x m
-- @'insertWithKey' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
--- insert @f key new_value old_value@.
+-- insert the pair @(key,f key new_value old_value)@.
+-- Note that the key passed to f is the same key passed to 'insertWithKey'.
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWithKey f kx x t
= case t of
-> case compare kx ky of
LT -> balance ky y (insertWithKey f kx x l) r
GT -> balance ky y l (insertWithKey f kx x r)
- EQ -> Bin sy ky (f ky x y) l r
+ EQ -> Bin sy kx (f kx x y) l r
-- | /O(log n)/. The expression (@'insertLookupWithKey' f k x map@)
-- is a pair where the first element is equal to (@'lookup' k map@)
-> case compare kx ky of
LT -> let (found,l') = insertLookupWithKey f kx x l in (found,balance ky y l' r)
GT -> let (found,r') = insertLookupWithKey f kx x r in (found,balance ky y l r')
- EQ -> (Just y, Bin sy ky (f ky x y) l r)
+ EQ -> (Just y, Bin sy kx (f kx x y) l r)
{--------------------------------------------------------------------
Deletion
-- It prefers @t1@ when duplicate keys are encountered,
-- i.e. (@'union' == 'unionWith' 'const'@).
-- The implementation uses the efficient /hedge-union/ algorithm.
--- Hedge-union is more efficient on (bigset `union` smallset)?
+-- Hedge-union is more efficient on (bigset `union` smallset)
union :: Ord k => Map k a -> Map k a -> Map k a
union Tip t2 = t2
union t1 Tip = t1
-union t1 t2
- | size t1 >= size t2 = hedgeUnionL (const LT) (const GT) t1 t2
- | otherwise = hedgeUnionR (const LT) (const GT) t2 t1
+union t1 t2 = hedgeUnionL (const LT) (const GT) t1 t2
-- left-biased hedge union
hedgeUnionL cmplo cmphi t1 Tip
(found,gt) = trimLookupLo kx cmphi t2
newx = case found of
Nothing -> x
- Just y -> y
+ Just (_,y) -> y
{--------------------------------------------------------------------
Union with a combining function
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWithKey f Tip t2 = t2
unionWithKey f t1 Tip = t1
-unionWithKey f t1 t2
- | size t1 >= size t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
- | otherwise = hedgeUnionWithKey flipf (const LT) (const GT) t2 t1
- where
- flipf k x y = f k y x
+unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
hedgeUnionWithKey f cmplo cmphi t1 Tip
= t1
(found,gt) = trimLookupLo kx cmphi t2
newx = case found of
Nothing -> x
- Just y -> f kx x y
+ Just (_,y) -> f kx x y
{--------------------------------------------------------------------
Difference
hedgeDiffWithKey f cmplo cmphi t (Bin _ kx x l r)
= case found of
Nothing -> merge tl tr
- Just y -> case f kx y x of
- Nothing -> merge tl tr
- Just z -> join kx z tl tr
+ Just (ky,y) ->
+ case f ky y x of
+ Nothing -> merge tl tr
+ Just z -> join ky z tl tr
where
cmpkx k = compare kx k
lt = trim cmplo cmpkx t
-- | /O(n+m)/. Intersection with a combining function.
-- Intersection is more efficient on (bigset `intersection` smallset)
+--intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
+--intersectionWithKey f Tip t = Tip
+--intersectionWithKey f t Tip = Tip
+--intersectionWithKey f t1 t2 = intersectWithKey f t1 t2
+--
+--intersectWithKey f Tip t = Tip
+--intersectWithKey f t Tip = Tip
+--intersectWithKey f t (Bin _ kx x l r)
+-- = case found of
+-- Nothing -> merge tl tr
+-- Just y -> join kx (f kx y x) tl tr
+-- where
+-- (lt,found,gt) = splitLookup kx t
+-- tl = intersectWithKey f lt l
+-- tr = intersectWithKey f gt r
+
+
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
intersectionWithKey f Tip t = Tip
intersectionWithKey f t Tip = Tip
-intersectionWithKey f t1 t2
- | size t1 >= size t2 = intersectWithKey f t1 t2
- | otherwise = intersectWithKey flipf t2 t1
- where
- flipf k x y = f k y x
-
-intersectWithKey f Tip t = Tip
-intersectWithKey f t Tip = Tip
-intersectWithKey f t (Bin _ kx x l r)
- = case found of
+intersectionWithKey f t1@(Bin s1 k1 x1 l1 r1) t2@(Bin s2 k2 x2 l2 r2) =
+ if s1 >= s2 then
+ let (lt,found,gt) = splitLookupWithKey k2 t1
+ tl = intersectionWithKey f lt l2
+ tr = intersectionWithKey f gt r2
+ in case found of
+ Just (k,x) -> join k (f k x x2) tl tr
+ Nothing -> merge tl tr
+ else let (lt,found,gt) = splitLookup k1 t2
+ tl = intersectionWithKey f l1 lt
+ tr = intersectionWithKey f r1 gt
+ in case found of
+ Just x -> join k1 (f k1 x1 x) tl tr
Nothing -> merge tl tr
- Just y -> join kx (f kx y x) tl tr
- where
- (lt,found,gt) = splitLookup kx t
- tl = intersectWithKey f lt l
- tr = intersectWithKey f gt r
le -> trim cmplo cmphi l
ge -> trim cmplo cmphi r
-trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe a, Map k a)
+trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a)
trimLookupLo lo cmphi Tip = (Nothing,Tip)
trimLookupLo lo cmphi t@(Bin sx kx x l r)
= case compare lo kx of
LT -> case cmphi kx of
- GT -> (lookup lo t, t)
+ GT -> (lookupAssoc lo t, t)
le -> trimLookupLo lo cmphi l
GT -> trimLookupLo lo cmphi r
- EQ -> (Just x,trim (compare lo) cmphi r)
+ EQ -> (Just (kx,x),trim (compare lo) cmphi r)
{--------------------------------------------------------------------
GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
EQ -> (l,Just x,r)
+-- | /O(log n)/.
+splitLookupWithKey :: Ord k => k -> Map k a -> (Map k a,Maybe (k,a),Map k a)
+splitLookupWithKey k Tip = (Tip,Nothing,Tip)
+splitLookupWithKey k (Bin sx kx x l r)
+ = case compare k kx of
+ LT -> let (lt,z,gt) = splitLookupWithKey k l in (lt,z,join kx x gt r)
+ GT -> let (lt,z,gt) = splitLookupWithKey k r in (join kx x l lt,z,gt)
+ EQ -> (l,Just (kx, x),r)
+
+-- | /O(log n)/. Performs a 'split' but also returns whether the pivot
+-- element was found in the original set.
+splitMember :: Ord k => k -> Map k a -> (Map k a,Bool,Map k a)
+splitMember x t = let (l,m,r) = splitLookup x t in
+ (l,maybe False (const True) m,r)
+
+
{--------------------------------------------------------------------
Utility functions that maintain the balance properties of the tree.
All constructors assume that all values in [l] < [k] and all values
-- SIAM journal of computing 2(1), March 1973.
--
-- Note that the implementation is /left-biased/ -- the elements of a
--- first argument are always perferred to the second, for example in
+-- first argument are always preferred to the second, for example in
-- 'union' or 'insert'. Of course, left-biasing can only be observed
-- when equality is an equivalence relation instead of structural
-- equality.
union :: Ord a => Set a -> Set a -> Set a
union Tip t2 = t2
union t1 Tip = t1
-union t1 t2
- | size t1 >= size t2 = hedgeUnion (const LT) (const GT) t1 t2
- | otherwise = hedgeUnion (const LT) (const GT) t2 t1
+union t1 t2 = hedgeUnion (const LT) (const GT) t1 t2
hedgeUnion cmplo cmphi t1 Tip
= t1
Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. The intersection of two sets.
--- Intersection is more efficient on (bigset `intersection` smallset).
+-- Elements of the result come from the first set.
intersection :: Ord a => Set a -> Set a -> Set a
intersection Tip t = Tip
intersection t Tip = Tip
-intersection t1 t2
- | size t1 >= size t2 = intersect' t1 t2
- | otherwise = intersect' t2 t1
-
-intersect' Tip t = Tip
-intersect' t Tip = Tip
-intersect' t (Bin _ x l r)
- | found = join x tl tr
- | otherwise = merge tl tr
- where
- (lt,found,gt) = splitMember x t
- tl = intersect' lt l
- tr = intersect' gt r
-
+intersection t1@(Bin s1 x1 l1 r1) t2@(Bin s2 x2 l2 r2) =
+ if s1 >= s2 then
+ let (lt,found,gt) = splitLookup x2 t1
+ tl = intersection lt l2
+ tr = intersection gt r2
+ in case found of
+ Just x -> join x tl tr
+ Nothing -> merge tl tr
+ else let (lt,found,gt) = splitMember x1 t2
+ tl = intersection l1 lt
+ tr = intersection r1 gt
+ in if found then join x1 tl tr
+ else merge tl tr
{--------------------------------------------------------------------
Filter and partition
-- | /O(log n)/. Performs a 'split' but also returns whether the pivot
-- element was found in the original set.
splitMember :: Ord a => a -> Set a -> (Set a,Bool,Set a)
-splitMember x Tip = (Tip,False,Tip)
-splitMember x (Bin sy y l r)
- = case compare x y of
- LT -> let (lt,found,gt) = splitMember x l in (lt,found,join y gt r)
- GT -> let (lt,found,gt) = splitMember x r in (join y l lt,found,gt)
- EQ -> (l,True,r)
+splitMember x t = let (l,m,r) = splitLookup x t in
+ (l,maybe False (const True) m,r)
+
+-- | /O(log n)/. Performs a 'split' but also returns the pivot
+-- element that was found in the original set.
+splitLookup :: Ord a => a -> Set a -> (Set a,Maybe a,Set a)
+splitLookup x Tip = (Tip,Nothing,Tip)
+splitLookup x (Bin sy y l r)
+ = case compare x y of
+ LT -> let (lt,found,gt) = splitLookup x l in (lt,found,join y gt r)
+ GT -> let (lt,found,gt) = splitLookup x r in (join y l lt,found,gt)
+ EQ -> (l,Just y,r)
{--------------------------------------------------------------------
Utility functions that maintain the balance properties of the tree.
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