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-- From http://www.reddit.com/r/haskell/comments/ti5il/redblack_trees_in_haskell_using_gadts_existential/
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DataKinds#-}
{-# LANGUAGE KindSignatures#-}
module RedBlackTree where
import Data.Kind (Type)
data Nat = Zero | Succ Nat deriving (Eq, Ord, Show)
type One = Succ Zero
data RedBlack = Black | Red deriving (Eq, Ord, Show)
-- red-black trees are rooted at a black node
data RedBlackTree a = forall n. T ( Node Black n a )
deriving instance Show a => Show (RedBlackTree a)
-- all paths from a node to a leaf have exactly n black nodes
data Node :: RedBlack -> Nat -> Type -> Type where
-- all leafs are black
Leaf :: Node Black One a
-- internal black nodes can have children of either color
B :: Node cL n a -> a -> Node cR n a -> Node Black (Succ n) a
-- internal red nodes can only have black children
R :: Node Black n a -> a -> Node Black n a -> Node Red n a
deriving instance Show a => Show (Node c n a)
-- one-hole context for red-black trees
data Context :: Nat -> RedBlack -> Nat -> Type -> Type where
-- if we're at the root, the hole is a black node
Root :: Context n Black n a
-- we can go left or right from a red node hole, creating a hole for a black node
BC :: Bool -> a -> Node Black n a -> Context m Red n a -> Context m Black n a
-- we can go left or right from a black node hole, creating a hole for either
EC :: Bool -> a -> Node cY n a -> Context m Black (Succ n) a -> Context m cX n a
deriving instance Show a => Show (Context m c n a)
data Zipper m a = forall c n. Zipper (Node c n a) (Context m c n a)
deriving instance Show a => Show (Zipper m a)
-- create a zipper
unZip :: Node Black n a -> Zipper n a
unZip = flip Zipper Root
-- destroy a zipper
zipUp :: Zipper m a -> Node Black m a
zipUp (Zipper x Root) = x
zipUp (Zipper x (BC goLeft a y c)) = zipUp $ Zipper (if goLeft then R x a y else R y a x) c
zipUp (Zipper x (EC goLeft a y c)) = zipUp $ Zipper (if goLeft then B x a y else B y a x) c
-- locate the node that should contain a in the red-black tree
zipTo :: Ord a => a -> Zipper n a -> Zipper n a
zipTo _ z@(Zipper Leaf _) = z
zipTo a z@(Zipper (R l a' r) c) = case compare a a' of
EQ -> z
LT -> zipTo a $ Zipper l (BC True a' r c)
GT -> zipTo a $ Zipper r (BC False a' l c)
zipTo a z@(Zipper (B l a' r) c) = case compare a a' of
EQ -> z
LT -> zipTo a $ Zipper l (EC True a' r c)
GT -> zipTo a $ Zipper r (EC False a' l c)
-- create a red-black tree
empty :: RedBlackTree a
empty = T Leaf
-- insert a node into a red-black tree
-- (see http://en.wikipedia.org/wiki/Red%E2%80%93black_tree#Insertion)
insert :: Ord a => a -> RedBlackTree a -> RedBlackTree a
insert a t@(T root) = case zipTo a (unZip root) of
-- find matching leaf and replace with red node (pointing to two leaves)
Zipper Leaf c -> insertAt (R Leaf a Leaf) c
-- if it's already in the tree, there's no need to modify it
_ -> t
insertAt :: Node Red n a -> Context m c n a -> RedBlackTree a
-- 1) new node is root => paint it black and done
insertAt (R l a r) Root = T $ B l a r
-- 2) new node's parent is black => done
insertAt x (EC b a y c) = T . zipUp $ Zipper x (EC b a y c)
-- 3) uncle is red => paint parent/uncle black, g'parent red. recurse on g'parent
insertAt x (BC pb pa py (EC gb ga (R ul ua ur) gc)) = insertAt g gc
where p = if pb then B x pa py else B py pa x
u = B ul ua ur
g = if gb then R p ga u else R u ga p
-- 4) node is between parent and g'parent => inner rotation
insertAt (R l a r) (BC False pa py pc@(EC True _ _ _)) = insertAt (R py pa l) (BC True a r pc)
insertAt (R l a r) (BC True pa py pc@(EC False _ _ _)) = insertAt (R r pa py) (BC False a l pc)
-- 5) otherwise => outer rotation
insertAt x (BC True pa py (EC True ga gy@Leaf gc)) =
T . zipUp $ Zipper (B x pa $ R py ga gy) gc
insertAt x (BC True pa py (EC True ga gy@(B _ _ _) gc)) =
T . zipUp $ Zipper (B x pa $ R py ga gy) gc
-- XXX: GHC seems unable to infer that gy is Black so I have to do both cases
-- explicitly, rather than
-- insertAt x (BC False pa py (EC False ga gy gc)) =
-- T . zipUp $ Zipper (B (R gy ga py) pa x) gc
--
-- insertAt x (BC True pa py (EC True ga gy gc)) =
-- T . zipUp $ Zipper (B x pa $ R py ga gy) gc
{-
BC :: Bool -> a -> Node Black n a -> Context m Red n a -> Context m Black n a
EC :: Bool -> a -> Node cY n a -> Context m Black (Succ n) a -> Context m cX n a
BC True pa py (EC True ga gy gc) :: Context m c n a
Hence c~Black
EC True ga gy gc :: Context m Red n a
gy :: Node cY n a
-}
insertAt x (BC False pa py (EC False ga gy@Leaf gc)) =
T . zipUp $ Zipper (B (R gy ga py) pa x) gc
insertAt x (BC False pa py (EC False ga gy@(B _ _ _) gc)) =
T . zipUp $ Zipper (B (R gy ga py) pa x) gc
-- can't derive, since we abstract over n, so we have to manually
-- check for identical structure
instance Eq a => Eq (RedBlackTree a) where
T Leaf == T Leaf = True
T (B l@(B _ _ _) a r@(B _ _ _)) == T (B l'@(B _ _ _) a' r'@(B _ _ _)) =
a == a' && T l == T l' && T r == T r'
T (B (R ll la lr) a r@(B _ _ _)) == T (B (R ll' la' lr') a' r'@(B _ _ _)) =
a == a' && la == la' &&
T ll == T ll' && T lr == T lr' && T r == T r'
T (B l@(B _ _ _) a r@(R rl ra rr)) == T (B l'@(B _ _ _) a' r'@(R rl' ra' rr')) =
a == a' && ra == ra' &&
T l == T l' && T rl == T rl' && T rr == T rr'
T (B (R ll la lr) a r@(R rl ra rr)) == T (B (R ll' la' lr') a' r'@(R rl' ra' rr')) =
a == a' && la == la' && ra == ra' &&
T ll == T ll' && T lr == T lr' && T rl == T rl' && T rr == T rr'
_ == _ = False
-- can't derive, since B abstracts over child node colors, so
-- manually check for identical structure
instance (Eq a) => Eq (Node c n a) where
Leaf == Leaf = True
R l a r == R l' a' r' = a == a' && l == l' && r == r'
b@(B _ _ _) == b'@(B _ _ _) = T b == T b'
_ == _ = False
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