{-|
An implementation of bidirectional maps between values of two
key types. A 'Bimap' is essentially a bijection between subsets of
its two argument types.

Each element of the left-hand type is associated with an element
of the right-hand type, and vice-versa, such that the two mappings
are inverses. Deleting an element will cause its twin to be deleted,
and inserting a pair of elements will cause any overlapping bindings
to be deleted.

Most functions implicitly consider the left-hand type to be the
key, and the right-hand type to be the value.
Functions with an @R@ suffix reverse this convention, treating the
right-hand type as the key and the left-hand type as the value.
-}
module Data.Bimap (
    -- * Bimap type
    Bimap(),
    -- * Query
    null,
    size,
    member,
    memberR,
    notMember,
    notMemberR,
    pairMember,
    pairNotMember,
    lookup,
    lookupR,
    (!),
    (!>),
    -- * Construction
    empty,
    singleton,
    -- * Update
    insert,
    tryInsert,
    delete,
    deleteR,
    -- * Min\/Max
    findMin,
    findMinR,
    findMax,
    findMaxR,
    deleteMin,
    deleteMinR,
    deleteMax,
    deleteMaxR,
    deleteFindMin,
    deleteFindMinR,
    deleteFindMax,
    deleteFindMaxR,
    -- * Filter
    filter,
    partition,
    -- * Conversion\/traversal
    fromList,
    fromAList,
    fromAscPairList,
    fromAscPairListUnchecked,
    toList,
    toAscList,
    toAscListR,
    keys,
    keysR,
    elems,
    assocs,
    fold,
    toMap,
    toMapR,
    -- * Miscellaneous
    valid,
    twist,
    twisted,
) where

import Data.List (foldl', sort)
import qualified Data.Map as M
import Prelude hiding (lookup, null, filter, pred)
import qualified Prelude as P


infixr 9 .:
(.:) :: (c -> d) -> (a -> b -> c) -> a -> b -> d
(.:) = (.).(.)

{-|
A bidirectional map between values of types @a@ and @b@.
-}
data Bimap a b = MkBimap !(M.Map a b) !(M.Map b a)

instance (Show a, Show b) => Show (Bimap a b) where
    show x = "fromList " ++ (show . toList $ x)

instance (Eq a, Eq b) => Eq (Bimap a b) where
    (==) bx by = toAscList bx == toAscList by

{-| /O(1)/. The empty bimap.
/Version: 0.2/-}
empty :: Bimap a b
empty = MkBimap M.empty M.empty

{-| /O(1)/. A bimap with a single element.
/Version: 0.2/-}
singleton :: a -> b -> Bimap a b
singleton x y = MkBimap (M.singleton x y) (M.singleton y x)

{-| /O(1)/. Is the bimap empty?
/Version: 0.2/-}
null :: Bimap a b -> Bool
null (MkBimap left _) = M.null left

{-| /O(1)/. The number of elements in the bimap.
/Version: 0.2/-}
size :: Bimap a b -> Int
size (MkBimap left _) = M.size left

{-| /O(log n)/. Is the specified value a member of the bimap?
/Version: 0.2/-}
member :: (Ord a, Ord b) => a -> Bimap a b -> Bool
member x (MkBimap left _) = M.member x left
{-| /O(log n)/. A version of 'member' specialized to the right key.
/Version: 0.2/-}
memberR :: (Ord a, Ord b) => b -> Bimap a b -> Bool
memberR y (MkBimap _ right) = M.member y right

{-| /O(log n)/. Is the specified value not a member of the bimap?
/Version: 0.2/-}
notMember :: (Ord a, Ord b) => a -> Bimap a b -> Bool
notMember = not .: member
{-| /O(log n)/. A version of 'notMember' specialized to the right key.
/Version: 0.2/-}
notMemberR :: (Ord a, Ord b) => b -> Bimap a b -> Bool
notMemberR = not .: memberR

{-| /O(log n)/.
Are the two values associated /with each other/ in the bimap?

This function is uncurried in its first two arguments, so that it
can be used infix.

/Version: 0.2/-}
pairMember :: (Ord a, Ord b)
           => (a, b) -> Bimap a b -> Bool
pairMember (x, y) (MkBimap left _) =
    maybe False (== y) (M.lookup x left)

{-| /O(log n)/.
Are the two values not in the bimap, or not associated
with each other? (Complement of 'pairMember'.)
/Version: 0.2/-}
pairNotMember :: (Ord a, Ord b)
              => (a, b) -> Bimap a b -> Bool
pairNotMember = not .: pairMember

{-| /O(log n)/.
Insert a pair of values into the bimap, associating them.

If either of the values is already in the bimap, any overlapping
bindings are deleted.

/Version: 0.2/-}
insert :: (Ord a, Ord b)
       => a -> b -> Bimap a b -> Bimap a b
insert x y = delete x >>> deleteR y >>> unsafeInsert x y
    where
    (>>>) = flip (.)

{-| /O(log n)/.
Insert a pair of values into the bimap, but only if neither is
already in the bimap.
/Version: 0.2.2/-}
tryInsert :: (Ord a, Ord b)
          => a -> b -> Bimap a b -> Bimap a b
tryInsert x y bi
    | (x `notMember` bi && y `notMemberR` bi) = unsafeInsert x y bi
    | otherwise                               = bi

{-| /O(log n)/.
Insert a pair of values into the bimap, without checking for
overlapping bindings.

If either value is already in the bimap, and
is not bound to the other value, the bimap will become inconsistent.
-}
unsafeInsert :: (Ord a, Ord b)
             => a -> b -> Bimap a b -> Bimap a b
unsafeInsert x y (MkBimap left right) =
    MkBimap (M.insert x y left) (M.insert y x right)

{-| /O(log n)/. Common implementation for 'delete' and 'deleteR'. -}
deleteE :: (Ord a, Ord b)
       => Either a b -> Bimap a b -> Bimap a b
deleteE e (MkBimap left right) =
    MkBimap
        (perhaps M.delete x $ left)
        (perhaps M.delete y $ right)
    where
    perhaps = maybe id
    x = either Just (flip M.lookup right) e
    y = either (flip M.lookup left) Just  e

{-| /O(log n)/.
Delete a value and its twin from a bimap.

When the value is not a member of the bimap, the original bimap is
returned.

/Version: 0.2/-}
delete :: (Ord a, Ord b) => a -> Bimap a b -> Bimap a b
delete = deleteE . Left

{-| /O(log n)/ A version of 'delete' specialized to the right key.
/Version: 0.2/-}
deleteR :: (Ord a, Ord b) => b -> Bimap a b -> Bimap a b
deleteR = deleteE . Right

{-| /O(log n)/.
Lookup a left key in the bimap, returning the associated right key.

This function will @return@ the result in the monad, or @fail@ if
the value isn't in the bimap.

/Version: 0.2/-}
lookup :: (Ord a, Ord b, Monad m)
        => a -> Bimap a b -> m b
lookup x (MkBimap left _) =
    maybe (fail "Data.Bimap.lookup: Left key not found")
          (return)
          (M.lookup x left)

{-| /O(log n)/.
A version of 'lookup' that is specialized to the right key,
and returns the corresponding left key.
/Version: 0.2/-}
lookupR :: (Ord a, Ord b, Monad m)
        => b -> Bimap a b -> m a
lookupR y (MkBimap _ right) =
    maybe (fail "Data.Bimap.lookupR: Right key not found")
          (return)
          (M.lookup y right)

{-| /O(log n)/.
Find the right key corresponding to a given left key.
Calls @'error'@ when the key is not in the bimap.
/Version: 0.2/-}
(!) :: (Ord a, Ord b) => Bimap a b -> a -> b
(!) bi x = case lookup x bi of
    Just y  -> y
    Nothing -> error "Data.Bimap.(!): Left key not found"

{-| /O(log n)/.
A version of @(!)@ that is specialized to the right key,
and returns the corresponding left key.
/Version: 0.2/-}
(!>) :: (Ord a, Ord b) => Bimap a b -> b -> a
(!>) bi y = case lookupR y bi of
    Just x  -> x
    Nothing -> error "Data.Bimap.(!>): Right key not found"

{-| /O(n*log n)/.
Build a map from a list of pairs. If there are any overlapping
pairs in the list, the later ones will override the earlier ones.
/Version: 0.2/-}
fromList :: (Ord a, Ord b)
         => [(a, b)] -> Bimap a b
fromList xs = foldl' (flip . uncurry $ insert) empty xs

{-| /O(n*log n)/.
Build a map from a list of pairs. Unlike 'fromList', earlier pairs
will take precedence over later ones.

The name @fromAList@ is a reference to Lisp-style association
lists, where associations can be overridden by prepending new ones.

Note that when duplicates occur in both the keys and in the values,
@fromList xs /= fromAList (reverse xs)@. However, if either
contains no duplicates, then the equality holds.

/Version: 0.2.2/-}
fromAList :: (Ord a, Ord b)
          => [(a, b)] -> Bimap a b
fromAList xs = foldl' (flip . uncurry $ tryInsert) empty xs

{-| /O(n)/. Convert to a list of associated pairs.
/Version: 0.2/-}
toList :: Bimap a b -> [(a, b)]
toList = toAscList

{-| /O(n)/. Build a bimap from a list of pairs, where both the @fst@
and @snd@ halves of the list are in strictly ascending order.

This precondition is checked; an invalid list will cause an error.

/Version: 0.2.3/-}
fromAscPairList :: (Ord a, Ord b)
                => [(a, b)] -> Bimap a b
fromAscPairList xs
    | isBiAscending xs = fromAscPairListUnchecked xs
    | otherwise        = error
        "Data.Bimap.fromAscPairList: list not correctly ascending"

isBiAscending :: (Ord a, Ord b)
              => [(a, b)] -> Bool
isBiAscending = allAdjacent bothLess
    where
    -- True if the binary relation f is true for all adjacent pairs
    -- in the input list
    allAdjacent :: (c -> c -> Bool) -> [c] -> Bool
    allAdjacent f xs = all (uncurry f) $ zip xs (tail xs)
    -- True if both components of the first pair are strictly less
    -- than their counterparts in the second pair
    bothLess (x1, y1) (x2, y2) = (x1 < x2) && (y1 < y2)

{-| /O(n)/. Build a bimap from a list of pairs, where both the @fst@
and @snd@ halves of the list are in strictly ascending order.

This precondition is /not/ checked; an invalid list will produce a
malformed bimap.

/Version: 0.2.3/-}
fromAscPairListUnchecked :: (Ord a, Ord b)
                         => [(a, b)] -> Bimap a b
fromAscPairListUnchecked xs = MkBimap
    (M.fromAscList xs)
    (M.fromAscList $ map swap  xs)
    where
    swap (x, y) = (y, x)

{-| /O(n)/.
Convert to a list of associated pairs, with the left-hand
values in ascending order.

Since pair ordering is lexical, the pairs will also be in
ascending order.

/Version: 0.2/-}
toAscList :: Bimap a b -> [(a, b)]
toAscList (MkBimap left _) = M.toList left

{-| /O(n)/.
Convert to a list of associated pairs, with the right-hand
values first in the pair and in ascending order.

Since pair ordering is lexical, the pairs will also be in
ascending order.

/Version: 0.2/-}
toAscListR :: Bimap a b -> [(b, a)]
toAscListR = toAscList . twist

{-| /O(n)/.
Return all associated pairs in the bimap, with the left-hand
values in ascending order.
/Version: 0.2/-}
assocs :: Bimap a b -> [(a, b)]
assocs = toList

{-| /O(n)/.
Return all left-hand keys in the bimap in ascending order.
/Version: 0.2/-}
keys :: Bimap a b -> [a]
keys (MkBimap left _) = M.keys left

{-| /O(n)/.
Return all right-hand keys in the bimap in ascending order.
/Version: 0.2/-}
keysR :: Bimap a b -> [b]
keysR (MkBimap _ right) = M.keys right

{-| /O(n)/. An alias for 'keysR'.
/Version: 0.2/-}
elems :: Bimap a b -> [b]
elems = keysR

{-| /O(1)/. Extract only the left-to-right component of a bimap.
/Version: 0.2.1/-}
toMap :: Bimap a b -> M.Map a b
toMap (MkBimap left _) = left

{-| /O(1)/. Extract only the right-to-left component of a bimap.
/Version: 0.2.1/-}
toMapR :: Bimap a b -> M.Map b a
toMapR (MkBimap _ right) = right

{-| /O(n)/.
Filter all association pairs that satisfy the predicate.

Note that the predicate will be applied /twice/ for each association
in the bimap.

/Version: 0.2.4/-}
filter :: (Ord a, Ord b)
              => (a -> b -> Bool) -> Bimap a b -> Bimap a b
filter pred (MkBimap left right) =
    MkBimap
        (M.filterWithKey pred left)
        (M.filterWithKey (flip pred) right)

{-| /O(n)/.
Partition the bimap according to a predicate.
The first bimap contains all associations that satisfy the predicate;
the second contains all associations that fail the predicate.

Note that the predicate will be applied /twice/ for each association
in the bimap.

/Version: 0.2.4/-}
partition :: (Ord a, Ord b)
          => (a -> b -> Bool) -> Bimap a b -> (Bimap a b, Bimap a b)
partition pred (MkBimap left right) =
    (,) (MkBimap leftA rightA) (MkBimap leftB rightB)
    where
    (leftA, leftB) = M.partitionWithKey pred left
    (rightA, rightB) = M.partitionWithKey (flip pred) right


{-| /O(n*log n)/.
Test if the internal bimap structure is valid. This should be true
for any bimap created using the public interface, unless
'fromAscPairListUnchecked' has been used inappropriately.
/Version: 0.2/-}
valid :: (Ord a, Ord b)
      => Bimap a b -> Bool
valid (MkBimap left right) = and
    [ M.valid left, M.valid right
    , (==)
        (sort .                M.toList $ left )
        (sort . map flipPair . M.toList $ right)
    ]
    where
    flipPair (x, y) = (y, x)

{-| /O(1)/.
Reverse the positions of the two element types in the bimap.
/Version: 0.2/-}
twist ::  Bimap a b -> Bimap b a
twist (MkBimap left right) = MkBimap right left

{-| /O(1)/.
Reverse the positions of the two element types in a bimap
transformation.
/Version: 0.2/-}
twisted :: (Bimap a b -> Bimap a b) -> (Bimap b a -> Bimap b a)
twisted f = twist . f . twist

{-| /O(n)/.
Fold the association pairs in the map, such that
@'fold' f z == 'foldr' f z . 'assocs'@.
/Version: 0.2/-}
fold :: (a -> b -> c -> c) -> c -> Bimap a b -> c
fold f z = foldr (uncurry f) z . assocs

{-| /O(log n)/.
Delete and find the element with maximal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteFindMax :: (Ord b) => Bimap a b -> ((a, b), Bimap a b)
deleteFindMax (MkBimap left right) = ((a, b), MkBimap left' right') where
    ((a, b), left') = M.deleteFindMax left
    right' = b `M.delete` right

{-| /O(log n)/.
Delete and find the element with maximal right key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteFindMaxR :: (Ord a) => Bimap a b ->  ((b, a), Bimap a b)
deleteFindMaxR = second twist . deleteFindMax . twist where
    second f (x, y) = (x, f y)

{-| /O(log n)/.
Delete the element with maximal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteMax :: (Ord b) => Bimap a b -> Bimap a b
deleteMax = snd . deleteFindMax
 
{-| /O(log n)/.
Delete the element with maximal right key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteMaxR :: (Ord a) => Bimap a b -> Bimap a b
deleteMaxR = snd . deleteFindMaxR

{-| /O(log n)/.
Find the element with maximal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
findMax :: Bimap a b -> (a, b)
findMax = M.findMax . toMap

{-| /O(log n)/.
Find the element with maximal right key. The 
right-hand key is the first entry in the pair.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
findMaxR :: Bimap a b -> (b, a)
findMaxR = M.findMax . toMapR

{-| /O(log n)/.
Delete and find the element with minimal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteFindMin :: (Ord b) => Bimap a b -> ((a, b), Bimap a b)
deleteFindMin (MkBimap left right) = ((a, b), MkBimap left' right') where
    ((a, b), left') = M.deleteFindMin left
    right' = b `M.delete` right

{-| /O(log n)/.
Delete and find the element with minimal right key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteFindMinR :: (Ord a) => Bimap a b ->  ((b, a), Bimap a b)
deleteFindMinR = second twist . deleteFindMin . twist where
    second f (x, y) = (x, f y)

{-| /O(log n)/.
Delete the element with minimal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteMin :: (Ord b) => Bimap a b -> Bimap a b
deleteMin = snd . deleteFindMin
 
{-| /O(log n)/.
Delete the element with minimal right key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
deleteMinR :: (Ord a) => Bimap a b -> Bimap a b
deleteMinR = snd . deleteFindMinR

{-| /O(log n)/.
Find the element with minimal left key.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
findMin :: Bimap a b -> (a, b)
findMin = M.findMin . toMap

{-| /O(log n)/.
Find the element with minimal right key. The 
right-hand key is the first entry in the pair.
Calls @'error'@ if the bimap is empty.
/Version: 0.2.2/-}
findMinR :: Bimap a b -> (b, a)
findMinR = M.findMin . toMapR