1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
|
-- 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
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 -> * -> * 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 -> * -> * 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
|
