```{-|
Purely functional left-leaning red-black trees.

* Robert Sedgewick, \"Left-Leaning Red-Black Trees\",
Data structures seminar at Dagstuhl, Feb 2008.
<http://www.cs.princeton.edu/~rs/talks/LLRB/LLRB.pdf>

* Robert Sedgewick, \"Left-Leaning Red-Black Trees\",
Analysis of Algorithms meeting at Maresias, Apr 2008
<http://www.cs.princeton.edu/~rs/talks/LLRB/RedBlack.pdf>
-}

module Data.RBTree.LL (
-- * Data structures
RBTree(..)
, Color(..)
, BlackHeight
-- * Creating red-black trees
, empty
, insert
, singleton
, fromList
-- * Converting a list
, toList
-- * Membership
, member
-- * Deleting
, delete
, deleteMin
, deleteMax
-- * Checking
, null
-- * Set operations
, union
, intersection
, difference
-- * Helper functions
, join
, merge
, split
, valid
, minimum
, maximum
, showTree
, printTree
) where

import Data.List (foldl')
import Prelude hiding (minimum, maximum, null)

----------------------------------------------------------------
-- Part to be shared
----------------------------------------------------------------

data RBTree a = Leaf -- color is Black
| Node Color !BlackHeight !(RBTree a) a !(RBTree a)
deriving (Show)

data Color = B -- ^ Black
| R -- ^ Red
deriving (Eq,Show)

{-|
Red nodes have the same BlackHeight of their parent.
-}
type BlackHeight = Int

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

instance (Eq a) => Eq (RBTree a) where
t1 == t2 = toList t1 == toList t2

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

height :: RBTree a -> BlackHeight
height Leaf = 0
height (Node _ h _ _ _) = h

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

{-|
See if the red black tree is empty.

>>> Data.RBTree.LL.null empty
True
>>> Data.RBTree.LL.null (singleton 1)
False
-}

null :: Eq a => RBTree a -> Bool
null t = t == Leaf

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

{-| Empty tree.

>>> height empty
0
-}

empty :: RBTree a
empty = Leaf

{-| Singleton tree.

>>> height (singleton 'a')
1
-}

singleton :: Ord a => a -> RBTree a
singleton x = Node B 1 Leaf x Leaf

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

{-| Creating a tree from a list. O(N log N)

>>> empty == fromList []
True
>>> singleton 'a' == fromList ['a']
True
>>> fromList [5,3,5] == fromList [5,3]
True
-}

fromList :: Ord a => [a] -> RBTree a
fromList = foldl' (flip insert) empty

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

{-| Creating a list from a tree. O(N)

>>> toList (fromList [5,3])
[3,5]
>>> toList empty
[]
-}

toList :: RBTree a -> [a]
toList t = inorder t []
where
inorder Leaf xs = xs
inorder (Node _ _ l x r) xs = inorder l (x : inorder r xs)

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

{-| Checking if this element is a member of a tree?

>>> member 5 (fromList [5,3])
True
>>> member 1 (fromList [5,3])
False
-}

member :: Ord a => a -> RBTree a -> Bool
member _ Leaf = False
member x (Node _ _ l y r) = case compare x y of
LT -> member x l
GT -> member x r
EQ -> True

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

isBalanced :: RBTree a -> Bool
isBalanced t = isBlackSame t && isRedSeparate t

isBlackSame :: RBTree a -> Bool
isBlackSame t = all (n==) ns
where
n:ns = blacks t

blacks :: RBTree a -> [Int]
blacks = blacks' 0
where
blacks' n Leaf = [n+1]
blacks' n (Node R _ l _ r) = blacks' n  l ++ blacks' n  r
blacks' n (Node B _ l _ r) = blacks' n' l ++ blacks' n' r
where
n' = n + 1

isRedSeparate :: RBTree a -> Bool
isRedSeparate = reds B

reds :: Color -> RBTree t -> Bool
reds _ Leaf = True
reds R (Node R _ _ _ _) = False
reds _ (Node c _ l _ r) = reds c l && reds c r

isOrdered :: Ord a => RBTree a -> Bool
isOrdered t = ordered \$ toList t
where
ordered [] = True
ordered [_] = True
ordered (x:y:xys) = x < y && ordered (y:xys)

blackHeight :: RBTree a -> Bool
blackHeight Leaf = True
blackHeight t@(Node B i _ _ _) = bh i t
where
bh n Leaf = n == 0
bh n (Node R h l _ r) = n == h' && bh n l && bh n r
where
h' = h - 1
bh n (Node B h l _ r) = n == h && bh n' l && bh n' r
where
n' = n - 1
blackHeight _ = error "blackHeight"

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

turnR :: RBTree a -> RBTree a
turnR Leaf             = error "turnR"
turnR (Node _ h l x r) = Node R h l x r

turnB :: RBTree a -> RBTree a
turnB Leaf           = error "turnB"
turnB (Node _ h l x r) = Node B h l x r

turnB' :: RBTree a -> RBTree a
turnB' Leaf             = Leaf
turnB' (Node _ h l x r) = Node B h l x r

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

minimum :: RBTree a -> a
minimum (Node _ _ Leaf x _) = x
minimum (Node _ _ l _ _)    = minimum l
minimum _                   = error "minimum"

maximum :: RBTree a -> a
maximum (Node _ _ _ x Leaf) = x
maximum (Node _ _ _ _ r)    = maximum r
maximum _                   = error "maximum"

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

showTree :: Show a => RBTree a -> String
showTree = showTree' ""

showTree' :: Show a => String -> RBTree a -> String
showTree' _ Leaf = "\n"
showTree' pref (Node k h l x r) = show k ++ " " ++ show x ++ " (" ++ show h ++ ")\n"
++ pref ++ "+ " ++ showTree' pref' l
++ pref ++ "+ " ++ showTree' pref' r
where
pref' = "  " ++ pref

printTree :: Show a => RBTree a -> IO ()
printTree = putStr . showTree

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

isRed :: RBTree a -> Bool
isRed (Node R _ _ _ _ ) = True
isRed _               = False

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

isBlackLeftBlack :: RBTree a -> Bool
isBlackLeftBlack (Node B _ Leaf _ _)             = True
isBlackLeftBlack (Node B _ (Node B _ _ _ _) _ _) = True
isBlackLeftBlack _                               = False

isBlackLeftRed :: RBTree a -> Bool
isBlackLeftRed (Node B _ (Node R _ _ _ _) _ _) = True
isBlackLeftRed _                               = False

----------------------------------------------------------------
-- Basic operations
----------------------------------------------------------------

valid :: Ord a => RBTree a -> Bool
valid t = isBalanced t && isLeftLean t && blackHeight t && isOrdered t

isLeftLean :: RBTree a -> Bool
isLeftLean Leaf = True
isLeftLean (Node B _ _ _ (Node R _ _ _ _)) = False -- right only and both!
isLeftLean (Node _ _ r _ l) = isLeftLean r && isLeftLean l

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

{-| Insertion. O(log N)

>>> insert 5 (fromList [5,3]) == fromList [3,5]
True
>>> insert 7 (fromList [5,3]) == fromList [3,5,7]
True
>>> insert 5 empty            == singleton 5
True
-}

insert :: Ord a => a -> RBTree a -> RBTree a
insert kx t = turnB (insert' kx t)

insert' :: Ord a => a -> RBTree a -> RBTree a
insert' kx Leaf = Node R 1 Leaf kx Leaf
insert' kx t@(Node c h l x r) = case compare kx x of
LT -> balanceL c h (insert' kx l) x r
GT -> balanceR c h l x (insert' kx r)
EQ -> t

balanceL :: Color -> BlackHeight -> RBTree a -> a -> RBTree a -> RBTree a
balanceL B h (Node R _ ll@(Node R _ _ _ _) lx lr) x r =
Node R (h+1) (turnB ll) lx (Node B h lr x r)
balanceL c h l x r = Node c h l x r

balanceR :: Color -> BlackHeight -> RBTree a -> a -> RBTree a -> RBTree a
balanceR B h l@(Node R _ _ _ _) x r@(Node R _ _ _ _) =
Node R (h+1) (turnB l) x (turnB r)
-- x is Black since Red eliminated by the case above
-- x is either Node or Leaf
balanceR c h l x (Node R rh rl rx rr) = Node c h (Node R rh l x rl) rx rr
balanceR c h l x r = Node c h l x r

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

{-| Deleting the minimum element. O(log N)

>>> deleteMin (fromList [5,3,7]) == fromList [5,7]
True
>>> deleteMin empty == empty
True
-}

deleteMin :: RBTree a -> RBTree a
deleteMin Leaf = empty
deleteMin t = case deleteMin' (turnR t) of
Leaf -> Leaf
s    -> turnB s

{-
This deleteMin' keeps an invariant: the target node is always red.

If the left child of the minimum node is Leaf, the right child
MUST be Leaf thanks to the invariants of LLRB.
-}

deleteMin' :: RBTree a -> RBTree a
deleteMin' (Node R _ Leaf _ Leaf) = Leaf -- deleting the minimum
deleteMin' t@(Node R h l x r)
-- Red
| isRed l      = Node R h (deleteMin' l) x r
-- Black-Black
| isBB && isBR = hardMin t
| isBB         = balanceR B (h-1) (deleteMin' (turnR l)) x (turnR r)
-- Black-Red
| otherwise    = Node R h (Node B lh (deleteMin' ll) lx lr) x r -- ll is Red
where
isBB = isBlackLeftBlack l
isBR = isBlackLeftRed r
Node B lh ll lx lr = l -- to skip Black
deleteMin' _ = error "deleteMin'"

-- Simplified but not keeping the invariant.
{-
deleteMin' :: RBTree a -> RBTree a
deleteMin' (Node R _ Leaf _ Leaf) = Leaf
deleteMin' t@(Node R h l x r)
| isBB && isBR = hardMin t
| isBB         = balanceR B (h-1) (deleteMin' (turnR l)) x (turnR r)
where
isBB = isBlackLeftBlack l
isBR = isBlackLeftRed r
deleteMin' (Node c h l x r) = Node c h (deleteMin' l) x r
deleteMin' _ = error "deleteMin'"
-}

{-
The hardest case. See slide 61 of:
http://www.cs.princeton.edu/~rs/talks/LLRB/RedBlack.pdf
-}

hardMin :: RBTree a -> RBTree a
hardMin (Node R h l x (Node B rh (Node R _ rll rlx rlr) rx rr))
= Node R h (Node B rh (deleteMin' (turnR l)) x rll)
rlx
(Node B rh rlr rx rr)
hardMin _ = error "hardMin"

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

{-| Deleting the maximum

>>> deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
True
>>> deleteMax empty == empty
True
-}

deleteMax :: RBTree a -> RBTree a
deleteMax Leaf = empty
deleteMax t = case deleteMax' (turnR t) of
Leaf -> Leaf
s    -> turnB s

{-
This deleteMax' keeps an invariant: the target node is always red.

If the right child of the minimum node is Leaf, the left child
is:

1) A Leaf -- we can delete it
2) A red node -- we can rotateR it and have 1).
-}

deleteMax' :: RBTree a -> RBTree a
deleteMax' (Node R _ Leaf _ Leaf) = Leaf -- deleting the maximum
deleteMax' t@(Node R h l x r)
| isRed l      = rotateR t
-- Black-Black
| isBB && isBR = hardMax t
| isBB         = balanceR B (h-1) (turnR l) x (deleteMax' (turnR r))
-- Black-Red
| otherwise    = Node R h l x (rotateR r)
where
isBB = isBlackLeftBlack r
isBR = isBlackLeftRed l
deleteMax' _ = error "deleteMax'"

-- Simplified but not keeping the invariant.
{-
deleteMax' :: RBTree a -> RBTree a
deleteMax' (Node R _ Leaf _ Leaf) = Leaf
deleteMax' t@(Node _ _ (Node R _ _ _ _) _ _) = rotateR t
deleteMax' t@(Node R h l x r)
| isBB && isBR = hardMax t
| isBB         = balanceR B (h-1) (turnR l) x (deleteMax' (turnR r))
where
isBB = isBlackLeftBlack r
isBR = isBlackLeftRed l
deleteMax' (Node R h l x r) = Node R h l x (deleteMax' r)
deleteMax' _ = error "deleteMax'"
-}

{-
rotateR ensures that the maximum node is in the form of (Node R Leaf _ Leaf).
-}

rotateR :: RBTree a -> RBTree a
rotateR (Node c h (Node R _ ll lx lr) x r) = balanceR c h ll lx (deleteMax' (Node R h lr x r))
rotateR _ = error "rorateR"

{-
The hardest case. See slide 56 of:
http://www.cs.princeton.edu/~rs/talks/LLRB/RedBlack.pdf
-}

hardMax :: RBTree a -> RBTree a
hardMax (Node R h (Node B lh ll@(Node R _ _ _ _ ) lx lr) x r)
= Node R h (turnB ll) lx (balanceR B lh lr x (deleteMax' (turnR r)))
hardMax _              = error "hardMax"

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

{-| Deleting this element from a tree. O(log N)

>>> delete 5 (fromList [5,3]) == singleton 3
True
>>> delete 7 (fromList [5,3]) == fromList [3,5]
True
>>> delete 5 empty                         == empty
True
-}

delete :: Ord a => a -> RBTree a -> RBTree a
delete _ Leaf = empty
delete kx t = case delete' kx (turnR t) of
Leaf -> Leaf
t'   -> turnB t'

delete' :: Ord a => a -> RBTree a -> RBTree a
delete' _ Leaf = Leaf
delete' kx (Node c h l x r) = case compare kx x of
LT -> deleteLT kx c h l x r
GT -> deleteGT kx c h l x r
EQ -> deleteEQ kx c h l x r

deleteLT :: Ord a => a -> Color -> BlackHeight -> RBTree a -> a -> RBTree a -> RBTree a
deleteLT kx R h l x r
| isBB && isBR = Node R h (Node B rh (delete' kx (turnR l)) x rll) rlx (Node B rh rlr rx rr)
| isBB         = balanceR B (h-1) (delete' kx (turnR l)) x (turnR r)
where
isBB = isBlackLeftBlack l
isBR = isBlackLeftRed r
Node B rh (Node R _ rll rlx rlr) rx rr = r
deleteLT kx c h l x r = Node c h (delete' kx l) x r

deleteGT :: Ord a => a -> Color -> BlackHeight -> RBTree a -> a -> RBTree a -> RBTree a
deleteGT kx c h (Node R _ ll lx lr) x r = balanceR c h ll lx (delete' kx (Node R h lr x r))
deleteGT kx R h l x r
| isBB && isBR = Node R h (turnB ll) lx (balanceR B lh lr x (delete' kx (turnR r)))
| isBB         = balanceR B (h-1) (turnR l) x (delete' kx (turnR r))
where
isBB = isBlackLeftBlack r
isBR = isBlackLeftRed l
Node B lh ll@(Node R _ _ _ _) lx lr = l
deleteGT kx R h l x r = Node R h l x (delete' kx r)
deleteGT _ _ _ _ _ _ = error "deleteGT"

deleteEQ :: Ord a => a -> Color -> BlackHeight -> RBTree a -> a -> RBTree a -> RBTree a
deleteEQ _ R _ Leaf _ Leaf = Leaf
deleteEQ kx c h (Node R _ ll lx lr) x r = balanceR c h ll lx (delete' kx (Node R h lr x r))
deleteEQ _ R h l _ r
| isBB && isBR = balanceR R h (turnB ll) lx (balanceR B lh lr m (deleteMin' (turnR r)))
| isBB         = balanceR B (h-1) (turnR l) m (deleteMin' (turnR r))
where
isBB = isBlackLeftBlack r
isBR = isBlackLeftRed l
Node B lh ll@(Node R _ _ _ _) lx lr = l
m = minimum r
deleteEQ _ R h l _ r@(Node B rh rl rx rr) = Node R h l m (Node B rh (deleteMin' rl) rx rr) -- rl is Red
where
m = minimum r
deleteEQ _ _ _ _ _ _ = error "deleteEQ"

----------------------------------------------------------------
-- Set operations
----------------------------------------------------------------

{-
Each element of t1 < g.
Each element of t2 > g.
Both t1 and t2 must be Black.
-}

join :: Ord a => RBTree a -> a -> RBTree a -> RBTree a
join Leaf g t2 = insert g t2
join t1 g Leaf = insert g t1
join t1 g t2 = case compare h1 h2 of
LT -> turnB \$ joinLT t1 g t2 h1
GT -> turnB \$ joinGT t1 g t2 h2
EQ -> Node B (h1+1) t1 g t2
where
h1 = height t1
h2 = height t2

-- The root of result must be red.
joinLT :: Ord a => RBTree a -> a -> RBTree a -> BlackHeight -> RBTree a
joinLT t1 g t2@(Node c h l x r) h1
| h == h1   = Node R (h+1) t1 g t2
| otherwise = balanceL c h (joinLT t1 g l h1) x r
joinLT _ _ _ _ = error "joinLT"

-- The root of result must be red.
joinGT :: Ord a => RBTree a -> a -> RBTree a -> BlackHeight -> RBTree a
joinGT t1@(Node c h l x r) g t2 h2
| h == h2   = Node R (h+1) t1 g t2
| otherwise = balanceR c h l x (joinGT r g t2 h2)
joinGT _ _ _ _ = error "joinGT"

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

{-
Each element of t1 < each element of t2
Both t1 and t2 must be Black.
-}

merge :: Ord a => RBTree a -> RBTree a -> RBTree a
merge Leaf t2 = t2
merge t1 Leaf = t1
merge t1 t2 = case compare h1 h2 of
LT -> turnB \$ mergeLT t1 t2 h1
GT -> turnB \$ mergeGT t1 t2 h2
EQ -> turnB \$ mergeEQ t1 t2
where
h1 = height t1
h2 = height t2

mergeLT :: Ord a => RBTree a -> RBTree a -> BlackHeight -> RBTree a
mergeLT t1 t2@(Node c h l x r) h1
| h == h1   = mergeEQ t1 t2
| otherwise = balanceL c h (mergeLT t1 l h1) x r
mergeLT _ _ _ = error "mergeLT"

mergeGT :: Ord a => RBTree a -> RBTree a -> BlackHeight -> RBTree a
mergeGT t1@(Node c h l x r) t2 h2
| h == h2   = mergeEQ t1 t2
| otherwise = balanceR c h l x (mergeGT r t2 h2)
mergeGT _ _ _ = error "mergeGT"

{-
Merging two trees whose heights are the same.
The root must be either
a red with height + 1
for
a black with height
-}

mergeEQ :: Ord a => RBTree a -> RBTree a -> RBTree a
mergeEQ Leaf Leaf = Leaf
mergeEQ t1@(Node _ h l x r) t2
| h == h2'  = Node R (h+1) t1 m t2'
| isRed l   = Node R (h+1) (turnB l) x (Node B h r m t2')
| otherwise = Node B h (turnR t1) m t2'
where
m  = minimum t2
t2' = deleteMin t2
h2' = height t2'
mergeEQ _ _ = error "mergeEQ"

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

split :: Ord a => a -> RBTree a -> (RBTree a, RBTree a)
split _ Leaf = (Leaf,Leaf)
split kx (Node _ _ l x r) = case compare kx x of
LT -> (lt, join gt x r) where (lt,gt) = split kx l
GT -> (join l x lt, gt) where (lt,gt) = split kx r
EQ -> (turnB' l, r)

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

{-| Creating a union tree from two trees. O(N + M)

>>> union (fromList [5,3]) (fromList [5,7]) == fromList [3,5,7]
True
-}

union :: Ord a => RBTree a -> RBTree a -> RBTree a
union t1 Leaf = t1 -- ensured Black thanks to split
union Leaf t2 = turnB' t2
union t1 (Node _ _ l x r) = join (union l' l) x (union r' r)
where
(l',r') = split x t1

{-| Creating a intersection tree from trees. O(N + N)

>>> intersection (fromList [5,3]) (fromList [5,7]) == singleton 5
True
-}

intersection :: Ord a => RBTree a -> RBTree a -> RBTree a
intersection Leaf _ = Leaf
intersection _ Leaf = Leaf
intersection t1 (Node _ _ l x r)
| member x t1 = join (intersection l' l) x (intersection r' r)
| otherwise   = merge (intersection l' l) (intersection r' r)
where
(l',r') = split x t1

{-| Creating a difference tree from trees. O(N + N)

>>> difference (fromList [5,3]) (fromList [5,7]) == singleton 3
True
-}

difference :: Ord a => RBTree a -> RBTree a -> RBTree a
difference Leaf _  = Leaf
difference t1 Leaf = t1 -- ensured Black thanks to split
difference t1 (Node _ _ l x r) = merge (difference l' l) (difference r' r)
where
(l',r') = split x t1
```