{-# OPTIONS -cpp #-}

-----------------------------------------------------------------------------

-- |

-- Module : Common.Lib.Map

-- Copyright : (c) Daan Leijen 2002

-- License : BSD-style

-- Maintainer : libraries@haskell.org

-- Stability : provisional

-- Portability : portable

--

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

--

-- This module is intended to be imported @qualified@, to avoid name

-- clashes with Prelude functions. eg.

--

-- > import Common.Lib.Map as Map

--

-- 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.

-----------------------------------------------------------------------------

module Common.Lib.Map (

-- * Map type

Map -- instance Eq,Show

-- * Operators

, (!), (\\)

-- * Query

, null

, size

, member

, lookup

, findWithDefault

-- * Construction

, empty

, singleton

-- ** Insertion

, insert

, insertWith, insertWithKey, insertLookupWithKey

-- ** Delete\/Update

, delete

, adjust

, adjustWithKey

, update

, updateWithKey

, updateLookupWithKey

-- * Combine

-- ** Union

, union

, unionWith

, unionWithKey

, unions

, unionsWith

-- ** Difference

, difference

, differenceWith

, differenceWithKey

-- ** Intersection

, intersection

, intersectionWith

, intersectionWithKey

-- * Traversal

-- ** Map

, map

, mapWithKey

, mapAccum

, mapAccumWithKey

, mapKeys

, mapKeysWith

, mapKeysMonotonic

-- ** Fold

, fold

, foldWithKey

-- * Conversion

, elems

, keys

, keysSet

, assocs

-- ** Lists

, toList

, fromList

, fromListWith

, fromListWithKey

-- ** Ordered lists

, toAscList

, fromAscList

, fromAscListWith

, fromAscListWithKey

, fromDistinctAscList

-- * Filter

, filter

, filterWithKey

, partition

, partitionWithKey

, split

, splitLookup

-- * Submap

, isSubmapOf, isSubmapOfBy

, isProperSubmapOf, isProperSubmapOfBy

-- * 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,foldr,foldl,null)

#if __GLASGOW_HASKELL__<=602

import qualified Common.Lib.Set as Set

import qualified Data.List as List

import Data.Typeable

{-

-- for quick check

import qualified Prelude

import qualified List

import Debug.QuickCheck

import List(nub,sort)

-}

{--------------------------------------------------------------------

Operators

--------------------------------------------------------------------}

infixl 9 !,\\ --

-- | /O(log n)/. Find the value at a key.

-- Calls 'error' when the element can not be found.

(!) :: Ord k => Map k a -> k -> a

m ! k = find k m

-- | /O(n+m)/. See 'difference'.

(\\) :: Ord k => Map k a -> Map k b -> Map k a

m1 \\ m2 = difference m1 m2

{--------------------------------------------------------------------

Size balanced trees.

--------------------------------------------------------------------}

-- | A Map from keys @k@ to values @a@.

data Map k a = Tip

| Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)

type Size = Int

{--------------------------------------------------------------------

Query

--------------------------------------------------------------------}

-- | /O(1)/. Is the map empty?

null :: Map k a -> Bool

null 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 at a key in the map.

lookup :: (Monad m,Ord k) => k -> Map k a -> m a

lookup k t = case lookup' k t of

Just x -> return x

Nothing -> fail "Common.Lib.Map.lookup: Key not found"

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 at 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 at 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)/. The empty map.

empty :: Map k a

empty

= Tip

-- | /O(1)/. A map with a single element.

singleton :: k -> a -> Map k a

singleton k x

= Bin 1 k x Tip Tip

{--------------------------------------------------------------------

Insertion

--------------------------------------------------------------------}

-- | /O(log n)/. Insert a new key and value in the map.

-- If the key is already present in the map, the associated value is

-- replaced with the supplied value, i.e. 'insert' is equivalent to

-- @'insertWith' 'const'@.

insert :: Ord k => k -> a -> Map k a -> Map k a

insert kx x t

= case t of

Tip -> singleton 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 -> singleton 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, singleton 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 (@'updateWithKey' 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 :: (Monad m,Ord k) => k -> Map k a -> m Int

lookupIndex k t = case lookup 0 t of

Nothing -> fail "Common.Lib.Map.lookupIndex: Key not found."

Just x -> return x

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 value at 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 value at 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 value at 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 value at 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' == 'Prelude.foldl' 'union' 'empty'@).

unions :: Ord k => [Map k a] -> Map k a

unions ts

= foldlStrict union empty ts

-- | The union of a list of maps, with a combining operation:

-- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).

unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a

unionsWith f ts

= foldlStrict (unionWith f) 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,

-- i.e. (@'union' == 'unionWith' 'const'@).

-- The implementation uses the efficient /hedge-union/ algorithm.

-- 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

-- 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.

-- Hedge-union is more efficient on (bigset `union` smallset).

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

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 b -> 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 -> b -> Maybe a) -> Map k a -> Map k b -> 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 -> b -> Maybe a) -> Map k a -> Map k b -> 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 b -> 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 -> b -> c) -> Map k a -> Map k b -> Map k c

intersectionWith f m1 m2

= intersectionWithKey (\k x y -> f x y) m1 m2

-- | /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

| 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

(lt,found,gt) = splitLookup kx t

tl = intersectWithKey f lt l

tr = intersectWithKey f gt r

{--------------------------------------------------------------------

Submap

--------------------------------------------------------------------}

-- | /O(n+m)/.

-- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).

isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool

isSubmapOf m1 m2

= isSubmapOfBy (==) m1 m2

{- | /O(n+m)/.

The expression (@'isSubmapOfBy' 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':

> isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])

> isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])

> isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])

But the following are all 'False':

> isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])

> isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])

> isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])

-}

isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool

isSubmapOfBy f t1 t2

= (size t1 <= size t2) && (submap' f t1 t2)

submap' f Tip t = True

submap' f t Tip = False

submap' f (Bin _ kx x l r) t

= case found of

Nothing -> False

Just y -> f x y && submap' f l lt && submap' f r gt

where

(lt,found,gt) = splitLookup kx t

-- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).

-- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).

isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool

isProperSubmapOf m1 m2

= isProperSubmapOfBy (==) m1 m2

{- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).

The expression (@'isProperSubmapOfBy' 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':

> isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])

> isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])

But the following are all 'False':

> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])

> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])

> isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])

-}

isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

isProperSubmapOfBy f t1 t2

= (size t1 < size t2) && (submap' 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 ascending order of keys.

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 ascending order of keys.

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 order of keys.

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 order of keys.

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')

-- | /O(n*log n)/.

-- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.

--

-- The size of the result may be smaller if @f@ maps two or more distinct

-- keys to the same new key. In this case the value at the smallest of

-- these keys is retained.

mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a

mapKeys = mapKeysWith (\x y->x)

-- | /O(n*log n)/.

-- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.

--

-- The size of the result may be smaller if @f@ maps two or more distinct

-- keys to the same new key. In this case the associated values will be

-- combined using @c@.

mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a

mapKeysWith c f = fromListWith c . List.map fFirst . toList

where fFirst (x,y) = (f x, y)

-- | /O(n)/.

-- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@

-- is strictly monotonic.

-- /The precondition is not checked./

-- Semi-formally, we have:

--

-- > and [x < y ==> f x < f y | x <- ls, y <- ls]

-- > ==> mapKeysMonotonic f s == mapKeys f s

-- > where ls = keys s

mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a

mapKeysMonotonic f Tip = Tip

mapKeysMonotonic f (Bin sz k x l r) =

Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)

{--------------------------------------------------------------------

Folds

--------------------------------------------------------------------}

-- | /O(n)/. Fold the values in the map, such that

-- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.

-- For example,

--

-- > elems map = fold (:) [] map

--

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 keys and values in the map, such that

-- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.

-- For example,

--

-- > keys map = foldWithKey (\k x ks -> k:ks) [] map

--

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 in the ascending order of their keys.

elems :: Map k a -> [a]

elems m

= [x | (k,x) <- assocs m]

-- | /O(n)/. Return all keys of the map in ascending order.

keys :: Map k a -> [k]

keys m

= [k | (k,x) <- assocs m]

-- | /O(n)/. The set of all keys of the map.

keysSet :: Map k a -> Set.Set k

keysSet m = Set.fromDistinctAscList (keys m)

-- | /O(n)/. Return all key\/value pairs in the map in ascending key order.

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.

-- /The precondition (input list is ascending) is not checked./

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.

-- /The precondition (input list is ascending) is not checked./

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.

-- /The precondition (input list is ascending) is not checked./

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.

-- /The precondition is not checked./

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 (singleton k1 x1) (singleton k3 x3)) (singleton 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@. Any key equal to @k@ is found in neither @map1@ nor @map2@.

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 -> (Map k a,Maybe a,Map k a)

splitLookup k Tip = (Tip,Nothing,Tip)

splitLookup k (Bin sx kx x l r)

= case compare k kx of

LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)

GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)

EQ -> (l,Just x,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 -> singleton kx x

Bin sz ky y l r

-> balance ky y l (insertMax kx x r)

insertMin kx x t

= case t of

Tip -> singleton 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 automatic 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)

{--------------------------------------------------------------------

Ord

--------------------------------------------------------------------}

instance (Ord k, Ord v) => Ord (Map k v) where

compare m1 m2 = compare (toAscList m1) (toAscList m2)

{--------------------------------------------------------------------

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> let t = fromDistinctAscList [(x,()) | x <- [1..5]]

> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t

> (4,())

> +--(2,())

> | +--(1,())

> | +--(3,())

> +--(5,())

>

> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t

> (4,())

> |

> +--(2,())

> | |

> | +--(1,())

> | |

> | +--(3,())

> |

> +--(5,())

>

> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t

> +--(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 == singleton 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 (singleton 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])])

-}

#else

import Data.Map

#endif