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Red Black Trees (with GADTs 1)

> {-# LANGUAGE GADTs #-}                   
> {-# LANGUAGE DataKinds #-}              
> {-# LANGUAGE TypeFamilies #-}            
> {-# LANGUAGE TemplateHaskell #-}       

This version of RedBlack trees demonstrates the use of GADTs to statically verify RedBlack tree invariants.

• In this version, the type system enforces that the root is black by using a "singleton" type.

The definitions below are the same as the RedBlack module from last week, except that

1. we use standalone deriving for Show & Foldable, and give an explicit instance of Eq
2. we use the alternative GADT syntax to define the Color & RBT datatypes
3. we only do the insert function (we won't have time to demonstrate deletion)
4. we've slightly refactored balance

Below, most of the code should be familiar.

In preparation for the demo, we'll include a few additional language features, for GADTs, using datatypes in kinds, for type-level functions (new) and to easily run all of the QuickCheck properties in the file.

> module RedBlackGADT1 where

We'll make the following standard library functions available for this implementation.

> import qualified Data.Foldable as Foldable
> import Data.Kind(Type)

And we'll use QuickCheck for testing.

> import Test.QuickCheck hiding (elements)

API preview

Our goal is to use red-black trees to implement a finite set data structure, with a similar interface to Java's SortedSet or Haskell's Data.Set.

This module defines the following API for finite sets:

   type RBT a  -- a red-black tree containing elements of type a

   empty         :: RBT a
   insert        :: Ord a => a -> RBT a -> RBT a
   member        :: Ord a => a -> RBT a -> Bool
   elements      :: RBT a -> [a] 

This interface specifies a persistent set of ordered elements. We can tell that the implementation is persistent just by looking at the types of the operations. In particular, the empty operation is not a function, it is just a set --- there is only one empty set. If we were allowed to mutate it, it wouldn't be empty any more. Furthermore, the insert and delete operations return a new set instead of modifying their argument.

Red-black trees

Here, again, are the invariants for red-black trees:

1. The empty nodes at the leaves are black.

2. The root is black.

3. From each node, every path to a leaf has the same number of black nodes.

4. Red nodes have black children.

Tree Structure

If it has been a while since you have seen red-black trees, refresh your memory.

A red-black tree is a binary search tree where every node is marked with a color (red R or black B). For brevity, we will abbreviate the standard tree constructors Empty and Branch as E and N.

> data Color where
>   Red :: Color
>   Black :: Color

A "singleton type" for colors

> data SColor c where
>    R :: SColor Red
>    B :: SColor Black

We need a generalized form of equality. We don't want to force the two types to be the same.

> (%==) :: SColor c1 -> SColor c2 -> Bool
> R %== R = True
> B %== B = True
> _ %== _ = False

A "colored" tree

> data T c a where
>    E :: T Black a
>    N :: SColor c -> T c1 a -> a -> T c2 a -> T c a

We define the RBT type by distinguishing the root of the tree.

> data RBT a where
>    Root :: T Black a -> RBT a 

Type class instances

> -- Show instances

> deriving instance Show Color
> deriving instance (Show (SColor c))
> deriving instance Show a => Show (T c a) 
> deriving instance Show a => Show (RBT a)

> -- Eq instances

> instance Eq Color where
>    Red == Red = True
>    Black == Black = True
>    _ == _ = False

> -- Foldable instances

> deriving instance Foldable (T c)
> deriving instance Foldable RBT

Simple operations

We can access all of the elements of the red-black tree with an inorder tree traversal, directly available from the Foldable instance.

> -- | List all of the elements of the finite set, in ascending order
> elements :: RBT a -> [a]
> elements = Foldable.toList

Note above that we did not derive the Eq instance in the definition of RBT. Instead, we will define two red-black trees to be equal when they contain the same elements.

> instance Eq a => Eq (RBT a) where
>    (==) :: Eq a => RBT a -> RBT a -> Bool
>    t1 == t2 = elements t1 == elements t2

Every tree has a color, determined by the following function.

> -- | access the color of the tree
> color :: T c a -> SColor c
> color (N c _ _ _) = c
> color E = B

We can also calculate the "black height" of a tree -- i.e. the number of black nodes from the root to every leaf. It is an invariant that this number is the same for every path in the tree, so we only need to look at one side.

> -- | calculate the black height of the tree
> blackHeight :: T c a -> Int
> blackHeight E = 1
> blackHeight (N c a _ _) = blackHeight a + (if c %== B then 1 else 0)

Implementation

Not every value of type RBT a is a valid red-black tree.

Red-black trees must, first of all, be binary search trees. That means that the data in the tree must be stored in order.

Furthermore, red-black trees must satisfy also the following four invariants about colors.

1. Empty trees are black

2. The root (i.e. the topmost node) of a nonempty tree is black

3. From each node, every path to an E has the same number of black nodes

4. Red nodes have black children

• The first invariant is true by definition of the color function above. The others we will have to maintain as we implement the various tree operations.

• Together, these invariants imply that every red-black tree is "approximately balanced", in the sense that the longest path to an E is no more than twice the length of the shortest.

• From this balance property, it follows that the member, insert and delete operations will run in O(log_2 n) time.

Sample Trees

Here are some example trees; only the first one below is actually a red-black tree. The others violate the invariants above in some way.

> good1 :: RBT Int
> good1 = Root $ N B (N B E 1 E) 2 (N B E 3 E)

Here is one with a red Root (violates invariant 2). We want this definition to be rejected by the type checker.

> -- bad1 :: RBT Int
> -- bad1  = Root $ N R (N B E 1 E) 2 (N B E 3 E)

Here's one that violates the black height requirement (invariant 3). We want this definition to be rejected by the type checker. But this version doesn't have enough type information to do that.

> bad2 :: RBT Int
> bad2  = Root $ N B (N R E 1 E) 2 (N B E 3 E)

Here's a red-black tree that has a red node with a red child (violates invariant 4). We want this definition to be rejected by the type checker. But this version doesn't have enough type information to do that.

> bad3  :: RBT Int
> bad3  = Root $ N B (N R (N R E 1 E) 2 (N R E 3 E)) 4 E

Here's a red-black tree that isn't a binary search tree (i.e. the values stored in the tree are not in strictly increasing order). We won't use GADTs to enforce this property.

> bad4 :: RBT Int
> bad4  = Root $ N B (N B E 1 E) 3 (N B E 2 E)

All sample trees, plus the empty tree for good measure.

> trees :: [(String, RBT Int)]
> trees = [("good1", good1),
>          ("bad4", bad4),
>          ("empty",empty)]

Checking validity for red-black trees

We can write QuickCheck properties for each of the invariants above.

First, let's can define when a red-black tree satisfies the binary search tree condition. There are several ways of stating this condition, some of which are more efficient to check than others. Hughes suggests using an O(n^2) operation, because it obviously captures the invariant.

Here, we'll use a linear-time operation, and leave it to you to convince yourself that it is equivalent [4].

> -- | A red-black tree is a BST if an inorder traversal is strictly ordered.
> isBST :: Ord a => RBT a -> Bool
> isBST = orderedBy (<) . elements

> -- | Are the elements in the list ordered by the provided operation?
> orderedBy :: (a -> a -> Bool) -> [a] -> Bool
> orderedBy op (x:y:xs) = x `op` y && orderedBy op (y:xs)
> orderedBy _ _ = True

Now we can also think about validity properties for the colors in the tree.

1. The empty tree is black. (This is trivial, nothing to do here.)

2. The root of the tree is black.

> isRootBlack :: RBT a -> Bool
> isRootBlack (Root t) = color t %== B

3. For all nodes in the tree, all downward paths from the node to E contain the same number of black nodes. (Define this yourself, making sure that your test passes for good1 and fails for bad2.)

> consistentBlackHeight :: RBT a -> Bool
> consistentBlackHeight (Root t) = aux t where
>    aux :: T c a -> Bool
>    aux (N _ a _ b) = blackHeight a == blackHeight b && aux a && aux b
>    aux E = True

4. All children of red nodes are black.

> noRedRed :: RBT a  -> Bool
> noRedRed (Root t) = aux t where
>   aux :: T c a -> Bool
>   aux (N R a _ b) = color a %== B && color b %== B && aux a && aux b
>   aux (N B a _ b) = aux a && aux b
>   aux E = True

We can combine the predicates together using the following definition:

> valid :: Ord a => RBT a -> Bool
> valid t = isRootBlack t && consistentBlackHeight t && noRedRed t && isBST t

Take a moment to try out the properties above on the sample trees by running the testProps function in ghci. The good trees should satisfy all of the properties, whereas the bad trees should fail at least one of them.

> testProps :: IO ()
> testProps = mapM_ checkTree trees  where
>    checkTree (name,tree) = do
>      putStrLn $ "******* Checking " ++ name ++ " *******"
>      quickCheck $ once (counterexample "RB2" $ isRootBlack tree)
>      quickCheck $ once (counterexample "RB3" $ consistentBlackHeight tree)
>      quickCheck $ once (counterexample "RB4" $ noRedRed tree)
>      quickCheck $ once (counterexample "BST" $ isBST tree)

For convenience, we can also create a single property that combines all four color invariants together along with the BST invariant. The counterexample function reports which part of the combined property fails.

We will specialize all of the QuickCheck properties that we define to red-black trees that only contain small integer values.

> type A = Small Int

> prop_Valid :: RBT A -> Property
> prop_Valid tree = counterexample "RB2" (isRootBlack tree) .&&.
>                   counterexample "RB3" (consistentBlackHeight tree) .&&.
>                   counterexample "RB4" (noRedRed tree) .&&.
>                   counterexample "BST" (isBST tree)

Arbitrary Instance

> instance (Ord a, Arbitrary a) => Arbitrary (RBT a)  where
> 
>    arbitrary          :: Gen (RBT a)
>    arbitrary          = foldr insert empty <$> (arbitrary :: Gen [a])

>    shrink             :: RBT a -> [RBT a]
>    shrink (Root E)    = []
>    shrink (Root (N _ l _ r)) = [hide l, hide r] where
>         hide :: T c a -> RBT a
>         hide E = Root E
>         hide (N c l v r) = Root (N B l v r)

Implementation

> blacken :: HT a -> RBT a
> -- blacken E = Root E
> blacken (HN _ l v r) = Root (N B l v r)

> empty :: RBT a
> empty = Root E 

> member :: Ord a => a -> RBT a -> Bool
> member x0 (Root t) = aux x0 t where
>   aux :: Ord a => a -> T c a -> Bool
>   aux x E = False
>   aux x (N _ a y b)
>     | x < y     = aux x a
>     | x > y     = aux x b
>     | otherwise = True

> insert :: Ord a => a -> RBT a -> RBT a
> insert x (Root t) = blacken (ins x t)

> data HT a where
>   HN :: SColor c1 -> T c2 a -> a -> T c3 a -> HT a

> ins :: Ord a => a -> T c a -> HT a
> ins x E = HN R E x E
> ins x s@(N c a y b)
>   | x < y     = balanceL c (ins x a) y b
>   | x > y     = balanceR c a y (ins x b)
>   | otherwise = HN c a y b

The original balance function looked like this:

> -- Original version of balance
> {-
> balance (N B (N R (N R a x b) y c) z d) = N R (N B a x b) y (N B c z d)
> balance (N B (N R a x (N R b y c)) z d) = N R (N B a x b) y (N B c z d)

> balance (N B a x (N R (N R b y c) z d)) = N R (N B a x b) y (N B c z d)
> balance (N B a x (N R b y (N R c z d))) = N R (N B a x b) y (N B c z d)
> balance t = t
> -}

The first two clauses handled cases where the left subtree was unbalanced as a result of an insertion, while the last two handle cases where a right-insertion has unbalanced the tree.

Here, we split this function in two to recognize that we have information from ins above. We know exactly where to look for the red/red violation! If we inserted on the left, then we should balance on the left. If we inserted on the right, then we should balance on the right.

> balanceL :: SColor c1 -> HT a -> a -> T c2 a -> HT a
> balanceL B (HN R (N R a x b) y c) z d = HN R (N B a x b) y (N B c z d)
> balanceL B (HN R a x (N R b y c)) z d = HN R (N B a x b) y (N B c z d)
> balanceL c (HN c1 a x b) z d          = HN c (N c1 a x b) z d

> balanceR :: SColor c1 -> T c2 a -> a -> HT a -> HT a
> balanceR B a x (HN R (N R b y c) z d) = HN R (N B a x b) y (N B c z d)
> balanceR B a x (HN R b y (N R c z d)) = HN R (N B a x b) y (N B c z d)
> balanceR c a x (HN c1 b y d)          = HN c a x (N c1 b y d)

This version is slightly more efficient than the previous version and will be easier for us to augment with more precise types.

What properties should we test with QuickCheck?

• Validity Testing

We already have defined prop_Valid which tests whether its argument is a valid red-black tree. When we use this with the Arbitrary instance that we defined above, we are testing if the empty tree is valid and if the insert function preserves this invariant.

However, we also need to make sure that our shrink operation preserves invariants.

> prop_ShrinkValid :: RBT A -> Property
> prop_ShrinkValid t = conjoin (map prop_Valid (shrink t))

• Metamorphic Testing

> prop_InsertEmpty :: A -> Bool
> prop_InsertEmpty x = elements (insert x empty) == [x]

> prop_InsertInsert :: A -> A -> RBT A -> Bool
> prop_InsertInsert x y t =
>   insert x (insert y t) == insert y (insert x t)

> prop_MemberEmpty :: A -> Bool
> prop_MemberEmpty x = not (member x empty)

> prop_MemberInsert :: A -> A -> RBT A -> Bool
> prop_MemberInsert k k0 t =
>   member k (insert k0 t) == (k == k0 || member k t)

Running QuickCheck

Using the TemplateHaskell extension, the following code below defines an operation that will invoke QuickCheck with all definitions that start with prop_ above. This code must come after all of the definitions above (and runTests is not in scope before this point).

> return []
> runTests :: IO Bool
> runTests = $quickCheckAll