--------------------------------------------------------------------------------
{-| Module      :  Map
    Copyright   :  (c) Daan Leijen 2002
    License     :  BSD-style

    Maintainer  :  daan@cs.uu.nl
    Stability   :  provisional
    Portability :  portable

  An efficient implementation of maps from keys to values (dictionaries). 

  1) The module exports some names that clash with the "Prelude" -- 'lookup', 'map', and 'filter'. 
      If you want to use "Map" unqualified, these functions should be hidden.

      > import Prelude hiding (lookup,map,filter)
      > import Map

      Another solution is to use qualified names. This is also the only way how
      a "Map", "Set", and "MultiSet" can be used within one module. 

      > import qualified Map
      >
      > ... Map.single "Paris" "France"

      Or, if you prefer a terse coding style:

      > import qualified Map as M
      >
      > ... M.single "Berlin" "Germany"

  2) The implementation of "Map" is based on /size balanced/ binary trees (or
     trees of /bounded balance/) as described by:

     * Stephen Adams, \"/Efficient sets: a balancing act/\", Journal of Functional
       Programming 3(4):553-562, October 1993, <http://www.swiss.ai.mit.edu/~adams/BB>.

     * J. Nievergelt and E.M. Reingold, \"/Binary search trees of bounded balance/\",
       SIAM journal of computing 2(1), March 1973.
     
  3) Another implementation of finite maps based on size balanced trees
      exists as "Data.FiniteMap" in the Ghc libraries. The good part about this library 
      is that it is highly tuned and thorougly tested. However, it is also fairly old, 
      uses @#ifdef@'s all  over the place and only supports the basic finite map operations. 
      The "Map" module overcomes some of these issues:
        
      * It tries to export a more complete and consistent set of operations, like
        'partition', 'adjust', 'mapAccum', 'elemAt' etc. 
      
      * It uses the efficient /hedge/ algorithm for both 'union' and 'difference'
        (a /hedge/ algorithm is not applicable to 'intersection').
      
      * It converts ordered lists in linear time ('fromAscList').  

      * It takes advantage of the module system with names like 'empty' instead of 'Data.FiniteMap.emptyFM'.
      
      * It sticks to portable Haskell, avoiding @#ifdef@'s and other magic.
-}
----------------------------------------------------------------------------------
module Map  ( 
            -- * Map type
              Map          -- instance Eq,Show

            -- * Operators
            , (!), (\\)

            -- * Query
            , isEmpty
            , size
            , member
            , lookup
            , find          
            , findWithDefault
            
            -- * Construction
            , empty
            , single

            -- ** Insertion
            , insert
            , insertWith, insertWithKey, insertLookupWithKey
            
            -- ** Delete\/Update
            , delete
            , adjust
            , adjustWithKey
            , update
            , updateWithKey
            , updateLookupWithKey

            -- * Combine

            -- ** Union
            , union         
            , unionWith          
            , unionWithKey
            , unions

            -- ** Difference
            , difference
            , differenceWith
            , differenceWithKey
            
            -- ** Intersection
            , intersection           
            , intersectionWith
            , intersectionWithKey

            -- * Traversal
            -- ** Map
            , map
            , mapWithKey
            , mapAccum
            , mapAccumWithKey
            
            -- ** Fold
            , fold
            , foldWithKey

            -- * Conversion
            , elems
            , keys
            , assocs
            
            -- ** Lists
            , toList
            , fromList
            , fromListWith
            , fromListWithKey

            -- ** Ordered lists
            , toAscList
            , fromAscList
            , fromAscListWith
            , fromAscListWithKey
            , fromDistinctAscList

            -- * Filter 
            , filter
            , filterWithKey
            , partition
            , partitionWithKey

            , split         
            , splitLookup   

            -- * Subset
            , subset, subsetBy
            , properSubset, properSubsetBy

            -- * Indexed 
            , lookupIndex
            , findIndex
            , elemAt
            , updateAt
            , deleteAt

            -- * Min\/Max
            , findMin
            , findMax
            , deleteMin
            , deleteMax
            , deleteFindMin
            , deleteFindMax
            , updateMin
            , updateMax
            , updateMinWithKey
            , updateMaxWithKey
            
            -- * Debugging
            , showTree
            , showTreeWith
            , valid
            ) where

import Prelude hiding (lookup,map,filter)


{-
-- for quick check
import qualified Prelude
import qualified List
import Debug.QuickCheck       
import List(nub,sort)    
-}

{--------------------------------------------------------------------
  Operators
--------------------------------------------------------------------}
infixl 9 !,\\

-- | /O(log n)/. See 'find'.
(!) :: Ord k => Map k a -> k -> a
m ! k    = find k m

-- | /O(n+m)/. See 'difference'.
(\\) :: Ord k => Map k a -> Map k a -> Map k a
m1 \\ m2 = difference m1 m2

{--------------------------------------------------------------------
  Size balanced trees.
--------------------------------------------------------------------}
-- | A Map from keys @k@ and values @a@. 
data Map k a  = Tip 
              | Bin !Size !k a !(Map k a) !(Map k a) 

type Size     = Int

{--------------------------------------------------------------------
  Query
--------------------------------------------------------------------}
-- | /O(1)/. Is the map empty?
isEmpty :: Map k a -> Bool
isEmpty t
  = case t of
      Tip             -> True
      Bin sz k x l r  -> False

-- | /O(1)/. The number of elements in the map.
size :: Map k a -> Int
size t
  = case t of
      Tip             -> 0
      Bin sz k x l r  -> sz


-- | /O(log n)/. Lookup the value of key in the map.
lookup :: Ord k => k -> Map k a -> Maybe a
lookup k t
  = case t of
      Tip -> Nothing
      Bin sz kx x l r
          -> case compare k kx of
               LT -> lookup k l
               GT -> lookup k r
               EQ -> Just x       

-- | /O(log n)/. Is the key a member of the map?
member :: Ord k => k -> Map k a -> Bool
member k m
  = case lookup k m of
      Nothing -> False
      Just x  -> True

-- | /O(log n)/. Find the value of a key. Calls @error@ when the element can not be found.
find :: Ord k => k -> Map k a -> a
find k m
  = case lookup k m of
      Nothing -> error "Map.find: element not in the map"
      Just x  -> x

-- | /O(log n)/. The expression @(findWithDefault def k map)@ returns the value of key @k@ or returns @def@ when
-- the key is not in the map.
findWithDefault :: Ord k => a -> k -> Map k a -> a
findWithDefault def k m
  = case lookup k m of
      Nothing -> def
      Just x  -> x



{--------------------------------------------------------------------
  Construction
--------------------------------------------------------------------}
-- | /O(1)/. Create an empty map.
empty :: Map k a
empty 
  = Tip

-- | /O(1)/. Create a map with a single element.
single :: k -> a -> Map k a
single k x  
  = Bin 1 k x Tip Tip

{--------------------------------------------------------------------
  Insertion
  [insert] is the inlined version of [insertWith (\k x y -> x)]
--------------------------------------------------------------------}
-- | /O(log n)/. Insert a new key and value in the map.
insert :: Ord k => k -> a -> Map k a -> Map k a
insert kx x t
  = case t of
      Tip -> single kx x
      Bin sz ky y l r
          -> case compare kx ky of
               LT -> balance ky y (insert kx x l) r
               GT -> balance ky y l (insert kx x r)
               EQ -> Bin sz kx x l r

-- | /O(log n)/. Insert with a combining function.
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

-- | /O(log n)/. Insert with a combining function.
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWithKey f kx x t
  = case t of
      Tip -> single kx x
      Bin sy ky y l r
          -> 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

-- | /O(log n)/. The expression (@insertLookupWithKey f k x map@) is a pair where
-- the first element is equal to (@lookup k map@) and the second element
-- equal to (@insertWithKey f k x map@).
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a,Map k a)
insertLookupWithKey f kx x t
  = case t of
      Tip -> (Nothing, single kx x)
      Bin sy ky y l r
          -> 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)

{--------------------------------------------------------------------
  Deletion
  [delete] is the inlined version of [deleteWith (\k x -> Nothing)]
--------------------------------------------------------------------}
-- | /O(log n)/. Delete a key and its value from the map. When the key is not
-- a member of the map, the original map is returned.
delete :: Ord k => k -> Map k a -> Map k a
delete k t
  = case t of
      Tip -> Tip
      Bin sx kx x l r 
          -> case compare k kx of
               LT -> balance kx x (delete k l) r
               GT -> balance kx x l (delete k r)
               EQ -> glue l r

-- | /O(log n)/. Adjust a value at a specific key. When the key is not
-- a member of the map, the original map is returned.
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
adjust f k m
  = adjustWithKey (\k x -> f x) k m

-- | /O(log n)/. Adjust a value at a specific key. When the key is not
-- a member of the map, the original map is returned.
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
adjustWithKey f k m
  = updateWithKey (\k x -> Just (f k x)) k m

-- | /O(log n)/. The expression (@update f k map@) updates the value @x@
-- at @k@ (if it is in the map). If (@f x@) is @Nothing@, the element is
-- deleted. If it is (@Just y@), the key @k@ is bound to the new value @y@.
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
update f k m
  = updateWithKey (\k x -> f x) k m

-- | /O(log n)/. The expression (@update f k map@) updates the value @x@
-- at @k@ (if it is in the map). If (@f k x@) is @Nothing@, the element is
-- deleted. If it is (@Just y@), the key @k@ is bound to the new value @y@.
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
updateWithKey f k t
  = case t of
      Tip -> Tip
      Bin sx kx x l r 
          -> case compare k kx of
               LT -> balance kx x (updateWithKey f k l) r
               GT -> balance kx x l (updateWithKey f k r)
               EQ -> case f kx x of
                       Just x' -> Bin sx kx x' l r
                       Nothing -> glue l r

-- | /O(log n)/. Lookup and update.
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
updateLookupWithKey f k t
  = case t of
      Tip -> (Nothing,Tip)
      Bin sx kx x l r 
          -> case compare k kx of
               LT -> let (found,l') = updateLookupWithKey f k l in (found,balance kx x l' r)
               GT -> let (found,r') = updateLookupWithKey f k r in (found,balance kx x l r') 
               EQ -> case f kx x of
                       Just x' -> (Just x',Bin sx kx x' l r)
                       Nothing -> (Just x,glue l r)

{--------------------------------------------------------------------
  Indexing
--------------------------------------------------------------------}
-- | /O(log n)/. Return the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
-- the key is not a 'member' of the map.
findIndex :: Ord k => k -> Map k a -> Int
findIndex k t
  = case lookupIndex k t of
      Nothing  -> error "Map.findIndex: element is not in the map"
      Just idx -> idx

-- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map. 
lookupIndex :: Ord k => k -> Map k a -> Maybe Int
lookupIndex k t
  = lookup 0 t
  where
    lookup idx Tip  = Nothing
    lookup idx (Bin _ kx x l r)
      = case compare k kx of
          LT -> lookup idx l
          GT -> lookup (idx + size l + 1) r 
          EQ -> Just (idx + size l)

-- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
-- invalid index is used.
elemAt :: Int -> Map k a -> (k,a)
elemAt i Tip = error "Map.elemAt: index out of range"
elemAt i (Bin _ kx x l r)
  = case compare i sizeL of
      LT -> elemAt i l
      GT -> elemAt (i-sizeL-1) r
      EQ -> (kx,x)
  where
    sizeL = size l

-- | /O(log n)/. Update the element at /index/. Calls 'error' when an
-- invalid index is used.
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
updateAt f i Tip  = error "Map.updateAt: index out of range"
updateAt f i (Bin sx kx x l r)
  = case compare i sizeL of
      LT -> updateAt f i l
      GT -> updateAt f (i-sizeL-1) r
      EQ -> case f kx x of
              Just x' -> Bin sx kx x' l r
              Nothing -> glue l r
  where
    sizeL = size l

-- | /O(log n)/. Delete the element at /index/. Defined as (@deleteAt i map = updateAt (\k x -> Nothing) i map@).
deleteAt :: Int -> Map k a -> Map k a
deleteAt i map
  = updateAt (\k x -> Nothing) i map


{--------------------------------------------------------------------
  Minimal, Maximal
--------------------------------------------------------------------}
-- | /O(log n)/. The minimal key of the map.
findMin :: Map k a -> (k,a)
findMin (Bin _ kx x Tip r)  = (kx,x)
findMin (Bin _ kx x l r)    = findMin l
findMin Tip                 = error "Map.findMin: empty tree has no minimal element"

-- | /O(log n)/. The maximal key of the map.
findMax :: Map k a -> (k,a)
findMax (Bin _ kx x l Tip)  = (kx,x)
findMax (Bin _ kx x l r)    = findMax r
findMax Tip                 = error "Map.findMax: empty tree has no maximal element"

-- | /O(log n)/. Delete the minimal key
deleteMin :: Map k a -> Map k a
deleteMin (Bin _ kx x Tip r)  = r
deleteMin (Bin _ kx x l r)    = balance kx x (deleteMin l) r
deleteMin Tip                 = Tip

-- | /O(log n)/. Delete the maximal key
deleteMax :: Map k a -> Map k a
deleteMax (Bin _ kx x l Tip)  = l
deleteMax (Bin _ kx x l r)    = balance kx x l (deleteMax r)
deleteMax Tip                 = Tip

-- | /O(log n)/. Update the minimal key
updateMin :: (a -> Maybe a) -> Map k a -> Map k a
updateMin f m
  = updateMinWithKey (\k x -> f x) m

-- | /O(log n)/. Update the maximal key
updateMax :: (a -> Maybe a) -> Map k a -> Map k a
updateMax f m
  = updateMaxWithKey (\k x -> f x) m


-- | /O(log n)/. Update the minimal key
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMinWithKey f t
  = case t of
      Bin sx kx x Tip r  -> case f kx x of
                              Nothing -> r
                              Just x' -> Bin sx kx x' Tip r
      Bin sx kx x l r    -> balance kx x (updateMinWithKey f l) r
      Tip                -> Tip

-- | /O(log n)/. Update the maximal key
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMaxWithKey f t
  = case t of
      Bin sx kx x l Tip  -> case f kx x of
                              Nothing -> l
                              Just x' -> Bin sx kx x' l Tip
      Bin sx kx x l r    -> balance kx x l (updateMaxWithKey f r)
      Tip                -> Tip


{--------------------------------------------------------------------
  Union. 
--------------------------------------------------------------------}
-- | The union of a list of maps: (@unions == foldl union empty@).
unions :: Ord k => [Map k a] -> Map k a
unions ts
  = foldlStrict union empty ts

-- | /O(n+m)/.
-- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@. 
-- It prefers @t1@ when duplicate keys are encountered, ie. (@union == unionWith const@).
-- The implementation uses the efficient /hedge-union/ algorithm.
union :: Ord k => Map k a -> Map k a -> Map k a
union Tip t2  = t2
union t1 Tip  = t1
union t1 t2  -- hedge-union is more efficient on (bigset `union` smallset)
   | size t1 >= size t2  = hedgeUnionL (const LT) (const GT) t1 t2
   | otherwise           = hedgeUnionR (const LT) (const GT) t2 t1

-- left-biased hedge union
hedgeUnionL cmplo cmphi t1 Tip 
  = t1
hedgeUnionL cmplo cmphi Tip (Bin _ kx x l r)
  = join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionL cmplo cmphi (Bin _ kx x l r) t2
  = join kx x (hedgeUnionL cmplo cmpkx l (trim cmplo cmpkx t2)) 
              (hedgeUnionL cmpkx cmphi r (trim cmpkx cmphi t2))
  where
    cmpkx k  = compare kx k

-- right-biased hedge union
hedgeUnionR cmplo cmphi t1 Tip 
  = t1
hedgeUnionR cmplo cmphi Tip (Bin _ kx x l r)
  = join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionR cmplo cmphi (Bin _ kx x l r) t2
  = join kx newx (hedgeUnionR cmplo cmpkx l lt) 
                 (hedgeUnionR cmpkx cmphi r gt)
  where
    cmpkx k     = compare kx k
    lt          = trim cmplo cmpkx t2
    (found,gt)  = trimLookupLo kx cmphi t2
    newx        = case found of
                    Nothing -> x
                    Just y  -> y

{--------------------------------------------------------------------
  Union with a combining function
--------------------------------------------------------------------}
-- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWith f m1 m2
  = unionWithKey (\k x y -> f x y) m1 m2

-- | /O(n+m)/.
-- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
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  -- hedge-union is more efficient on (bigset `union` smallset)
  | 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

hedgeUnionWithKey f cmplo cmphi t1 Tip 
  = t1
hedgeUnionWithKey f cmplo cmphi Tip (Bin _ kx x l r)
  = join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionWithKey f cmplo cmphi (Bin _ kx x l r) t2
  = join kx newx (hedgeUnionWithKey f cmplo cmpkx l lt) 
                 (hedgeUnionWithKey f cmpkx cmphi r gt)
  where
    cmpkx k     = compare kx k
    lt          = trim cmplo cmpkx t2
    (found,gt)  = trimLookupLo kx cmphi t2
    newx        = case found of
                    Nothing -> x
                    Just y  -> f kx x y

{--------------------------------------------------------------------
  Difference
--------------------------------------------------------------------}
-- | /O(n+m)/. Difference of two maps. 
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
difference :: Ord k => Map k a -> Map k a -> Map k a
difference Tip t2  = Tip
difference t1 Tip  = t1
difference t1 t2   = hedgeDiff (const LT) (const GT) t1 t2

hedgeDiff cmplo cmphi Tip t     
  = Tip
hedgeDiff cmplo cmphi (Bin _ kx x l r) Tip 
  = join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeDiff cmplo cmphi t (Bin _ kx x l r) 
  = merge (hedgeDiff cmplo cmpkx (trim cmplo cmpkx t) l) 
          (hedgeDiff cmpkx cmphi (trim cmpkx cmphi t) r)
  where
    cmpkx k = compare kx k   

-- | /O(n+m)/. Difference with a combining function. 
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
differenceWith :: Ord k => (a -> a -> Maybe a) -> Map k a -> Map k a -> Map k a
differenceWith f m1 m2
  = differenceWithKey (\k x y -> f x y) m1 m2

-- | /O(n+m)/. Difference with a combining function. When two equal keys are
-- encountered, the combining function is applied to the key and both values.
-- If it returns @Nothing@, the element is discarded (proper set difference). If
-- it returns (@Just y@), the element is updated with a new value @y@. 
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
differenceWithKey :: Ord k => (k -> a -> a -> Maybe a) -> Map k a -> Map k a -> Map k a
differenceWithKey f Tip t2  = Tip
differenceWithKey f t1 Tip  = t1
differenceWithKey f t1 t2   = hedgeDiffWithKey f (const LT) (const GT) t1 t2

hedgeDiffWithKey f cmplo cmphi Tip t     
  = Tip
hedgeDiffWithKey f cmplo cmphi (Bin _ kx x l r) Tip 
  = join kx x (filterGt cmplo l) (filterLt cmphi r)
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
  where
    cmpkx k     = compare kx k   
    lt          = trim cmplo cmpkx t
    (found,gt)  = trimLookupLo kx cmphi t
    tl          = hedgeDiffWithKey f cmplo cmpkx lt l
    tr          = hedgeDiffWithKey f cmpkx cmphi gt r



{--------------------------------------------------------------------
  Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. Intersection of two maps. The values in the first
-- map are returned, i.e. (@intersection m1 m2 == intersectionWith const m1 m2@).
intersection :: Ord k => Map k a -> Map k a -> Map k a
intersection m1 m2
  = intersectionWithKey (\k x y -> x) m1 m2

-- | /O(n+m)/. Intersection with a combining function.
intersectionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
intersectionWith f m1 m2
  = intersectionWithKey (\k x y -> f x y) m1 m2

-- | /O(n+m)/. Intersection with a combining function.
intersectionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
intersectionWithKey f Tip t = Tip
intersectionWithKey f t Tip = Tip
intersectionWithKey f t1 t2  -- intersection is more efficient on (bigset `intersection` smallset)
  | 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
      Nothing -> merge tl tr
      Just y  -> join kx (f kx y x) tl tr
  where
    (found,lt,gt) = splitLookup kx t
    tl            = intersectWithKey f lt l
    tr            = intersectWithKey f gt r



{--------------------------------------------------------------------
  Subset
--------------------------------------------------------------------}
-- | /O(n+m)/. 
-- This function is defined as (@subset = subsetBy (==)@).
subset :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
subset m1 m2
  = subsetBy (==) m1 m2

{- | /O(n+m)/. 
 The expression (@subsetBy f t1 t2@) returns @True@ if
 all keys in @t1@ are in tree @t2@, and when @f@ returns @True@ when
 applied to their respective values. For example, the following 
 expressions are all @True@.
 
 > subsetBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > subsetBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > subsetBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])

 But the following are all @False@:
 
 > subsetBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
 > subsetBy (<)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > subsetBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
-}
subsetBy :: Ord k => (a->a->Bool) -> Map k a -> Map k a -> Bool
subsetBy f t1 t2
  = (size t1 <= size t2) && (subset' f t1 t2)

subset' f Tip t = True
subset' f t Tip = False
subset' f (Bin _ kx x l r) t
  = case found of
      Nothing -> False
      Just y  -> f x y && subset' f l lt && subset' f r gt
  where
    (found,lt,gt) = splitLookup kx t

-- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal). 
-- Defined as (@properSubset = properSubsetBy (==)@).
properSubset :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
properSubset m1 m2
  = properSubsetBy (==) m1 m2

{- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
 The expression (@properSubsetBy f m1 m2@) returns @True@ when
 @m1@ and @m2@ are not equal,
 all keys in @m1@ are in @m2@, and when @f@ returns @True@ when
 applied to their respective values. For example, the following 
 expressions are all @True@.
 
  > properSubsetBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
  > properSubsetBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])

 But the following are all @False@:
 
  > properSubsetBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
  > properSubsetBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
  > properSubsetBy (<)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
-}
properSubsetBy :: (Ord k,Eq a) => (a -> a -> Bool) -> Map k a -> Map k a -> Bool
properSubsetBy f t1 t2
  = (size t1 < size t2) && (subset' f t1 t2)

{--------------------------------------------------------------------
  Filter and partition
--------------------------------------------------------------------}
-- | /O(n)/. Filter all values that satisfy the predicate.
filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
filter p m
  = filterWithKey (\k x -> p x) m

-- | /O(n)/. Filter all keys\values that satisfy the predicate.
filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
filterWithKey p Tip = Tip
filterWithKey p (Bin _ kx x l r)
  | p kx x    = join kx x (filterWithKey p l) (filterWithKey p r)
  | otherwise = merge (filterWithKey p l) (filterWithKey p r)


-- | /O(n)/. partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a)
partition p m
  = partitionWithKey (\k x -> p x) m

-- | /O(n)/. partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
partitionWithKey p Tip = (Tip,Tip)
partitionWithKey p (Bin _ kx x l r)
  | p kx x    = (join kx x l1 r1,merge l2 r2)
  | otherwise = (merge l1 r1,join kx x l2 r2)
  where
    (l1,l2) = partitionWithKey p l
    (r1,r2) = partitionWithKey p r


{--------------------------------------------------------------------
  Mapping
--------------------------------------------------------------------}
-- | /O(n)/. Map a function over all values in the map.
map :: (a -> b) -> Map k a -> Map k b
map f m
  = mapWithKey (\k x -> f x) m

-- | /O(n)/. Map a function over all values in the map.
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
mapWithKey f Tip = Tip
mapWithKey f (Bin sx kx x l r) 
  = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)

-- | /O(n)/. The function @mapAccum@ threads an accumulating
-- argument through the map in an unspecified order.
mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccum f a m
  = mapAccumWithKey (\a k x -> f a x) a m

-- | /O(n)/. The function @mapAccumWithKey@ threads an accumulating
-- argument through the map in unspecified order. (= ascending pre-order)
mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumWithKey f a t
  = mapAccumL f a t

-- | /O(n)/. The function @mapAccumL@ threads an accumulating
-- argument throught the map in (ascending) pre-order.
mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumL f a t
  = case t of
      Tip -> (a,Tip)
      Bin sx kx x l r
          -> let (a1,l') = mapAccumL f a l
                 (a2,x') = f a1 kx x
                 (a3,r') = mapAccumL f a2 r
             in (a3,Bin sx kx x' l' r')

-- | /O(n)/. The function @mapAccumR@ threads an accumulating
-- argument throught the map in (descending) post-order.
mapAccumR :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumR f a t
  = case t of
      Tip -> (a,Tip)
      Bin sx kx x l r 
          -> let (a1,r') = mapAccumR f a r
                 (a2,x') = f a1 kx x
                 (a3,l') = mapAccumR f a2 l
             in (a3,Bin sx kx x' l' r')

{--------------------------------------------------------------------
  Folds  
--------------------------------------------------------------------}
-- | /O(n)/. Fold the map in an unspecified order. (= descending post-order).
fold :: (a -> b -> b) -> b -> Map k a -> b
fold f z m
  = foldWithKey (\k x z -> f x z) z m

-- | /O(n)/. Fold the map in an unspecified order. (= descending post-order).
foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
foldWithKey f z t
  = foldR f z t

-- | /O(n)/. In-order fold.
foldI :: (k -> a -> b -> b -> b) -> b -> Map k a -> b 
foldI f z Tip               = z
foldI f z (Bin _ kx x l r)  = f kx x (foldI f z l) (foldI f z r)

-- | /O(n)/. Post-order fold.
foldR :: (k -> a -> b -> b) -> b -> Map k a -> b
foldR f z Tip              = z
foldR f z (Bin _ kx x l r) = foldR f (f kx x (foldR f z r)) l

-- | /O(n)/. Pre-order fold.
foldL :: (b -> k -> a -> b) -> b -> Map k a -> b
foldL f z Tip              = z
foldL f z (Bin _ kx x l r) = foldL f (f (foldL f z l) kx x) r

{--------------------------------------------------------------------
  List variations 
--------------------------------------------------------------------}
-- | /O(n)/. Return all elements of the map.
elems :: Map k a -> [a]
elems m
  = [x | (k,x) <- assocs m]

-- | /O(n)/. Return all keys of the map.
keys  :: Map k a -> [k]
keys m
  = [k | (k,x) <- assocs m]

-- | /O(n)/. Return all key\/value pairs in the map.
assocs :: Map k a -> [(k,a)]
assocs m
  = toList m

{--------------------------------------------------------------------
  Lists 
  use [foldlStrict] to reduce demand on the control-stack
--------------------------------------------------------------------}
-- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
fromList :: Ord k => [(k,a)] -> Map k a 
fromList xs       
  = foldlStrict ins empty xs
  where
    ins t (k,x) = insert k x t

-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a 
fromListWith f xs
  = fromListWithKey (\k x y -> f x y) xs

-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a 
fromListWithKey f xs 
  = foldlStrict ins empty xs
  where
    ins t (k,x) = insertWithKey f k x t

-- | /O(n)/. Convert to a list of key\/value pairs.
toList :: Map k a -> [(k,a)]
toList t      = toAscList t

-- | /O(n)/. Convert to an ascending list.
toAscList :: Map k a -> [(k,a)]
toAscList t   = foldR (\k x xs -> (k,x):xs) [] t

-- | /O(n)/. 
toDescList :: Map k a -> [(k,a)]
toDescList t  = foldL (\xs k x -> (k,x):xs) [] t


{--------------------------------------------------------------------
  Building trees from ascending/descending lists can be done in linear time.
  
  Note that if [xs] is ascending that: 
    fromAscList xs       == fromList xs
    fromAscListWith f xs == fromListWith f xs
--------------------------------------------------------------------}
-- | /O(n)/. Build a map from an ascending list in linear time.
fromAscList :: Eq k => [(k,a)] -> Map k a 
fromAscList xs
  = fromAscListWithKey (\k x y -> x) xs

-- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a 
fromAscListWith f xs
  = fromAscListWithKey (\k x y -> f x y) xs

-- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a 
fromAscListWithKey f xs
  = fromDistinctAscList (combineEq f xs)
  where
  -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
  combineEq f xs
    = case xs of
        []     -> []
        [x]    -> [x]
        (x:xx) -> combineEq' x xx

  combineEq' z [] = [z]
  combineEq' z@(kz,zz) (x@(kx,xx):xs)
    | kx==kz    = let yy = f kx xx zz in combineEq' (kx,yy) xs
    | otherwise = z:combineEq' x xs


-- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
fromDistinctAscList :: [(k,a)] -> Map k a 
fromDistinctAscList xs
  = build const (length xs) xs
  where
    -- 1) use continutations so that we use heap space instead of stack space.
    -- 2) special case for n==5 to build bushier trees. 
    build c 0 xs   = c Tip xs 
    build c 5 xs   = case xs of
                       ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx) 
                            -> c (bin k4 x4 (bin k2 x2 (single k1 x1) (single k3 x3)) (single k5 x5)) xx
    build c n xs   = seq nr $ build (buildR nr c) nl xs
                   where
                     nl = n `div` 2
                     nr = n - nl - 1

    buildR n c l ((k,x):ys) = build (buildB l k x c) n ys
    buildB l k x c r zs     = c (bin k x l r) zs
                      


{--------------------------------------------------------------------
  Utility functions that return sub-ranges of the original
  tree. Some functions take a comparison function as argument to
  allow comparisons against infinite values. A function [cmplo k]
  should be read as [compare lo k].

  [trim cmplo cmphi t]  A tree that is either empty or where [cmplo k == LT]
                        and [cmphi k == GT] for the key [k] of the root.
  [filterGt cmp t]      A tree where for all keys [k]. [cmp k == LT]
  [filterLt cmp t]      A tree where for all keys [k]. [cmp k == GT]

  [split k t]           Returns two trees [l] and [r] where all keys
                        in [l] are <[k] and all keys in [r] are >[k].
  [splitLookup k t]     Just like [split] but also returns whether [k]
                        was found in the tree.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
  [trim lo hi t] trims away all subtrees that surely contain no
  values between the range [lo] to [hi]. The returned tree is either
  empty or the key of the root is between @lo@ and @hi@.
--------------------------------------------------------------------}
trim :: (k -> Ordering) -> (k -> Ordering) -> Map k a -> Map k a
trim cmplo cmphi Tip = Tip
trim cmplo cmphi t@(Bin sx kx x l r)
  = case cmplo kx of
      LT -> case cmphi kx of
              GT -> t
              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 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)
              le -> trimLookupLo lo cmphi l
      GT -> trimLookupLo lo cmphi r
      EQ -> (Just x,trim (compare lo) cmphi r)


{--------------------------------------------------------------------
  [filterGt k t] filter all keys >[k] from tree [t]
  [filterLt k t] filter all keys <[k] from tree [t]
--------------------------------------------------------------------}
filterGt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
filterGt cmp Tip = Tip
filterGt cmp (Bin sx kx x l r)
  = case cmp kx of
      LT -> join kx x (filterGt cmp l) r
      GT -> filterGt cmp r
      EQ -> r
      
filterLt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
filterLt cmp Tip = Tip
filterLt cmp (Bin sx kx x l r)
  = case cmp kx of
      LT -> filterLt cmp l
      GT -> join kx x l (filterLt cmp r)
      EQ -> l

{--------------------------------------------------------------------
  Split
--------------------------------------------------------------------}
-- | /O(log n)/. The expression (@split k map@) is a pair @(map1,map2)@ where
-- the keys in @map1@ are smaller than @k@ and the keys in @map2@ larger than @k@.
split :: Ord k => k -> Map k a -> (Map k a,Map k a)
split k Tip = (Tip,Tip)
split k (Bin sx kx x l r)
  = case compare k kx of
      LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
      GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
      EQ -> (l,r)

-- | /O(log n)/. The expression (@splitLookup k map@) splits a map just
-- like 'split' but also returns @lookup k map@.
splitLookup :: Ord k => k -> Map k a -> (Maybe a,Map k a,Map k a)
splitLookup k Tip = (Nothing,Tip,Tip)
splitLookup k (Bin sx kx x l r)
  = case compare k kx of
      LT -> let (z,lt,gt) = splitLookup k l in (z,lt,join kx x gt r)
      GT -> let (z,lt,gt) = splitLookup k r in (z,join kx x l lt,gt)
      EQ -> (Just x,l,r)

{--------------------------------------------------------------------
  Utility functions that maintain the balance properties of the tree.
  All constructors assume that all values in [l] < [k] and all values
  in [r] > [k], and that [l] and [r] are valid trees.
  
  In order of sophistication:
    [Bin sz k x l r]  The type constructor.
    [bin k x l r]     Maintains the correct size, assumes that both [l]
                      and [r] are balanced with respect to each other.
    [balance k x l r] Restores the balance and size.
                      Assumes that the original tree was balanced and
                      that [l] or [r] has changed by at most one element.
    [join k x l r]    Restores balance and size. 

  Furthermore, we can construct a new tree from two trees. Both operations
  assume that all values in [l] < all values in [r] and that [l] and [r]
  are valid:
    [glue l r]        Glues [l] and [r] together. Assumes that [l] and
                      [r] are already balanced with respect to each other.
    [merge l r]       Merges two trees and restores balance.

  Note: in contrast to Adam's paper, we use (<=) comparisons instead
  of (<) comparisons in [join], [merge] and [balance]. 
  Quickcheck (on [difference]) showed that this was necessary in order 
  to maintain the invariants. It is quite unsatisfactory that I haven't 
  been able to find out why this is actually the case! Fortunately, it 
  doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
  Join 
--------------------------------------------------------------------}
join :: Ord k => k -> a -> Map k a -> Map k a -> Map k a
join kx x Tip r  = insertMin kx x r
join kx x l Tip  = insertMax kx x l
join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
  | delta*sizeL <= sizeR  = balance kz z (join kx x l lz) rz
  | delta*sizeR <= sizeL  = balance ky y ly (join kx x ry r)
  | otherwise             = bin kx x l r


-- insertMin and insertMax don't perform potentially expensive comparisons.
insertMax,insertMin :: k -> a -> Map k a -> Map k a 
insertMax kx x t
  = case t of
      Tip -> single kx x
      Bin sz ky y l r
          -> balance ky y l (insertMax kx x r)
             
insertMin kx x t
  = case t of
      Tip -> single kx x
      Bin sz ky y l r
          -> balance ky y (insertMin kx x l) r
             
{--------------------------------------------------------------------
  [merge l r]: merges two trees.
--------------------------------------------------------------------}
merge :: Map k a -> Map k a -> Map k a
merge Tip r   = r
merge l Tip   = l
merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
  | delta*sizeL <= sizeR = balance ky y (merge l ly) ry
  | delta*sizeR <= sizeL = balance kx x lx (merge rx r)
  | otherwise            = glue l r

{--------------------------------------------------------------------
  [glue l r]: glues two trees together.
  Assumes that [l] and [r] are already balanced with respect to each other.
--------------------------------------------------------------------}
glue :: Map k a -> Map k a -> Map k a
glue Tip r = r
glue l Tip = l
glue l r   
  | size l > size r = let ((km,m),l') = deleteFindMax l in balance km m l' r
  | otherwise       = let ((km,m),r') = deleteFindMin r in balance km m l r'


-- | /O(log n)/. Delete and find the minimal element.
deleteFindMin :: Map k a -> ((k,a),Map k a)
deleteFindMin t 
  = case t of
      Bin _ k x Tip r -> ((k,x),r)
      Bin _ k x l r   -> let (km,l') = deleteFindMin l in (km,balance k x l' r)
      Tip             -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)

-- | /O(log n)/. Delete and find the maximal element.
deleteFindMax :: Map k a -> ((k,a),Map k a)
deleteFindMax t
  = case t of
      Bin _ k x l Tip -> ((k,x),l)
      Bin _ k x l r   -> let (km,r') = deleteFindMax r in (km,balance k x l r')
      Tip             -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)


{--------------------------------------------------------------------
  [balance l x r] balances two trees with value x.
  The sizes of the trees should balance after decreasing the
  size of one of them. (a rotation).

  [delta] is the maximal relative difference between the sizes of
          two trees, it corresponds with the [w] in Adams' paper.
  [ratio] is the ratio between an outer and inner sibling of the
          heavier subtree in an unbalanced setting. It determines
          whether a double or single rotation should be performed
          to restore balance. It is correspondes with the inverse
          of $\alpha$ in Adam's article.

  Note that:
  - [delta] should be larger than 4.646 with a [ratio] of 2.
  - [delta] should be larger than 3.745 with a [ratio] of 1.534.
  
  - A lower [delta] leads to a more 'perfectly' balanced tree.
  - A higher [delta] performs less rebalancing.

  - Balancing is automaic for random data and a balancing
    scheme is only necessary to avoid pathological worst cases.
    Almost any choice will do, and in practice, a rather large
    [delta] may perform better than smaller one.

  Note: in contrast to Adam's paper, we use a ratio of (at least) [2]
  to decide whether a single or double rotation is needed. Allthough
  he actually proves that this ratio is needed to maintain the
  invariants, his implementation uses an invalid ratio of [1].
--------------------------------------------------------------------}
delta,ratio :: Int
delta = 5
ratio = 2

balance :: k -> a -> Map k a -> Map k a -> Map k a
balance k x l r
  | sizeL + sizeR <= 1    = Bin sizeX k x l r
  | sizeR >= delta*sizeL  = rotateL k x l r
  | sizeL >= delta*sizeR  = rotateR k x l r
  | otherwise             = Bin sizeX k x l r
  where
    sizeL = size l
    sizeR = size r
    sizeX = sizeL + sizeR + 1

-- rotate
rotateL k x l r@(Bin _ _ _ ly ry)
  | size ly < ratio*size ry = singleL k x l r
  | otherwise               = doubleL k x l r

rotateR k x l@(Bin _ _ _ ly ry) r
  | size ry < ratio*size ly = singleR k x l r
  | otherwise               = doubleR k x l r

-- basic rotations
singleL k1 x1 t1 (Bin _ k2 x2 t2 t3)  = bin k2 x2 (bin k1 x1 t1 t2) t3
singleR k1 x1 (Bin _ k2 x2 t1 t2) t3  = bin k2 x2 t1 (bin k1 x1 t2 t3)

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)
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)


{--------------------------------------------------------------------
  The bin constructor maintains the size of the tree
--------------------------------------------------------------------}
bin :: k -> a -> Map k a -> Map k a -> Map k a
bin k x l r
  = Bin (size l + size r + 1) k x l r


{--------------------------------------------------------------------
  Eq converts the tree to a list. In a lazy setting, this 
  actually seems one of the faster methods to compare two trees 
  and it is certainly the simplest :-)
--------------------------------------------------------------------}
instance (Eq k,Eq a) => Eq (Map k a) where
  t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)

{--------------------------------------------------------------------
  Functor
--------------------------------------------------------------------}
instance Functor (Map k) where
  fmap f m  = map f m

{--------------------------------------------------------------------
  Show
--------------------------------------------------------------------}
instance (Show k, Show a) => Show (Map k a) where
  showsPrec d m  = showMap (toAscList m)

showMap :: (Show k,Show a) => [(k,a)] -> ShowS
showMap []     
  = showString "{}" 
showMap (x:xs) 
  = showChar '{' . showElem x . showTail xs
  where
    showTail []     = showChar '}'
    showTail (x:xs) = showChar ',' . showElem x . showTail xs
    
    showElem (k,x)  = shows k . showString ":=" . shows x
  

-- | /O(n)/. Show the tree that implements the map. The tree is shown
-- in a compressed, hanging format.
showTree :: (Show k,Show a) => Map k a -> String
showTree m
  = showTreeWith showElem True False m
  where
    showElem k x  = show k ++ ":=" ++ show x


{- | /O(n)/. The expression (@showTreeWith showelem hang wide map@) shows
 the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
 @wide@ is true, an extra wide version is shown.

>  Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False $ fromDistinctAscList [(x,()) | x <- [1..5]]
>  (4,())
>  +--(2,())
>  |  +--(1,())
>  |  +--(3,())
>  +--(5,())
>
>  Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True $ fromDistinctAscList [(x,()) | x <- [1..5]]
>  (4,())
>  |
>  +--(2,())
>  |  |
>  |  +--(1,())
>  |  |
>  |  +--(3,())
>  |
>  +--(5,())
>
>  Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True $ fromDistinctAscList [(x,()) | x <- [1..5]]
>  +--(5,())
>  |
>  (4,())
>  |
>  |  +--(3,())
>  |  |
>  +--(2,())
>     |
>     +--(1,())

-}
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
showTreeWith showelem hang wide t
  | hang      = (showsTreeHang showelem wide [] t) ""
  | otherwise = (showsTree showelem wide [] [] t) ""

showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
showsTree showelem wide lbars rbars t
  = case t of
      Tip -> showsBars lbars . showString "|\n"
      Bin sz kx x Tip Tip
          -> showsBars lbars . showString (showelem kx x) . showString "\n" 
      Bin sz kx x l r
          -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
             showWide wide rbars .
             showsBars lbars . showString (showelem kx x) . showString "\n" .
             showWide wide lbars .
             showsTree showelem wide (withEmpty lbars) (withBar lbars) l

showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
showsTreeHang showelem wide bars t
  = case t of
      Tip -> showsBars bars . showString "|\n" 
      Bin sz kx x Tip Tip
          -> showsBars bars . showString (showelem kx x) . showString "\n" 
      Bin sz kx x l r
          -> showsBars bars . showString (showelem kx x) . showString "\n" . 
             showWide wide bars .
             showsTreeHang showelem wide (withBar bars) l .
             showWide wide bars .
             showsTreeHang showelem wide (withEmpty bars) r


showWide wide bars 
  | wide      = showString (concat (reverse bars)) . showString "|\n" 
  | otherwise = id

showsBars :: [String] -> ShowS
showsBars bars
  = case bars of
      [] -> id
      _  -> showString (concat (reverse (tail bars))) . showString node

node           = "+--"
withBar bars   = "|  ":bars
withEmpty bars = "   ":bars


{--------------------------------------------------------------------
  Assertions
--------------------------------------------------------------------}
-- | /O(n)/. Test if the internal map structure is valid.
valid :: Ord k => Map k a -> Bool
valid t
  = balanced t && ordered t && validsize t

ordered t
  = bounded (const True) (const True) t
  where
    bounded lo hi t
      = case t of
          Tip              -> True
          Bin sz kx x l r  -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r

-- | Exported only for "Debug.QuickCheck"
balanced :: Map k a -> Bool
balanced t
  = case t of
      Tip              -> True
      Bin sz kx x l r  -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
                          balanced l && balanced r


validsize t
  = (realsize t == Just (size t))
  where
    realsize t
      = case t of
          Tip             -> Just 0
          Bin sz kx x l r -> case (realsize l,realsize r) of
                              (Just n,Just m)  | n+m+1 == sz  -> Just sz
                              other            -> Nothing

{--------------------------------------------------------------------
  Utilities
--------------------------------------------------------------------}
foldlStrict f z xs
  = case xs of
      []     -> z
      (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)


{-
{--------------------------------------------------------------------
  Testing
--------------------------------------------------------------------}
testTree xs   = fromList [(x,"*") | x <- xs]
test1 = testTree [1..20]
test2 = testTree [30,29..10]
test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]

{--------------------------------------------------------------------
  QuickCheck
--------------------------------------------------------------------}
qcheck prop
  = check config prop
  where
    config = Config
      { configMaxTest = 500
      , configMaxFail = 5000
      , configSize    = \n -> (div n 2 + 3)
      , configEvery   = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
      }


{--------------------------------------------------------------------
  Arbitrary, reasonably balanced trees
--------------------------------------------------------------------}
instance (Enum k,Arbitrary a) => Arbitrary (Map k a) where
  arbitrary = sized (arbtree 0 maxkey)
            where maxkey  = 10000

arbtree :: (Enum k,Arbitrary a) => Int -> Int -> Int -> Gen (Map k a)
arbtree lo hi n
  | n <= 0        = return Tip
  | lo >= hi      = return Tip
  | otherwise     = do{ x  <- arbitrary 
                      ; i  <- choose (lo,hi)
                      ; m  <- choose (1,30)
                      ; let (ml,mr)  | m==(1::Int)= (1,2)
                                     | m==2       = (2,1)
                                     | m==3       = (1,1)
                                     | otherwise  = (2,2)
                      ; l  <- arbtree lo (i-1) (n `div` ml)
                      ; r  <- arbtree (i+1) hi (n `div` mr)
                      ; return (bin (toEnum i) x l r)
                      }  


{--------------------------------------------------------------------
  Valid tree's
--------------------------------------------------------------------}
forValid :: (Show k,Enum k,Show a,Arbitrary a,Testable b) => (Map k a -> b) -> Property
forValid f
  = forAll arbitrary $ \t -> 
--    classify (balanced t) "balanced" $
    classify (size t == 0) "empty" $
    classify (size t > 0  && size t <= 10) "small" $
    classify (size t > 10 && size t <= 64) "medium" $
    classify (size t > 64) "large" $
    balanced t ==> f t

forValidIntTree :: Testable a => (Map Int Int -> a) -> Property
forValidIntTree f
  = forValid f

forValidUnitTree :: Testable a => (Map Int () -> a) -> Property
forValidUnitTree f
  = forValid f


prop_Valid 
  = forValidUnitTree $ \t -> valid t

{--------------------------------------------------------------------
  Single, Insert, Delete
--------------------------------------------------------------------}
prop_Single :: Int -> Int -> Bool
prop_Single k x
  = (insert k x empty == single k x)

prop_InsertValid :: Int -> Property
prop_InsertValid k
  = forValidUnitTree $ \t -> valid (insert k () t)

prop_InsertDelete :: Int -> Map Int () -> Property
prop_InsertDelete k t
  = (lookup k t == Nothing) ==> delete k (insert k () t) == t

prop_DeleteValid :: Int -> Property
prop_DeleteValid k
  = forValidUnitTree $ \t -> 
    valid (delete k (insert k () t))

{--------------------------------------------------------------------
  Balance
--------------------------------------------------------------------}
prop_Join :: Int -> Property 
prop_Join k 
  = forValidUnitTree $ \t ->
    let (l,r) = split k t
    in valid (join k () l r)

prop_Merge :: Int -> Property 
prop_Merge k
  = forValidUnitTree $ \t ->
    let (l,r) = split k t
    in valid (merge l r)


{--------------------------------------------------------------------
  Union
--------------------------------------------------------------------}
prop_UnionValid :: Property
prop_UnionValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (union t1 t2)

prop_UnionInsert :: Int -> Int -> Map Int Int -> Bool
prop_UnionInsert k x t
  = union (single k x) t == insert k x t

prop_UnionAssoc :: Map Int Int -> Map Int Int -> Map Int Int -> Bool
prop_UnionAssoc t1 t2 t3
  = union t1 (union t2 t3) == union (union t1 t2) t3

prop_UnionComm :: Map Int Int -> Map Int Int -> Bool
prop_UnionComm t1 t2
  = (union t1 t2 == unionWith (\x y -> y) t2 t1)

prop_UnionWithValid 
  = forValidIntTree $ \t1 ->
    forValidIntTree $ \t2 ->
    valid (unionWithKey (\k x y -> x+y) t1 t2)

prop_UnionWith :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_UnionWith xs ys
  = sum (elems (unionWith (+) (fromListWith (+) xs) (fromListWith (+) ys))) 
    == (sum (Prelude.map snd xs) + sum (Prelude.map snd ys))

prop_DiffValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (difference t1 t2)

prop_Diff :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_Diff xs ys
  =  List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys))) 
    == List.sort ((List.\\) (nub (Prelude.map fst xs))  (nub (Prelude.map fst ys)))

prop_IntValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (intersection t1 t2)

prop_Int :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_Int xs ys
  =  List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys))) 
    == List.sort (nub ((List.intersect) (Prelude.map fst xs)  (Prelude.map fst ys)))

{--------------------------------------------------------------------
  Lists
--------------------------------------------------------------------}
prop_Ordered
  = forAll (choose (5,100)) $ \n ->
    let xs = [(x,()) | x <- [0..n::Int]] 
    in fromAscList xs == fromList xs

prop_List :: [Int] -> Bool
prop_List xs
  = (sort (nub xs) == [x | (x,()) <- toList (fromList [(x,()) | x <- xs])])
-}