# Persistent Data Structures

Starring: Red-Black trees

`{-# LANGUAGE KindSignatures, ScopedTypeVariables #-}`

`module Persistent where`

```
import Control.Monad
import Test.QuickCheck hiding (elements)
import Data.Maybe as Maybe
import Data.List (sort,nub)
```

## Persistent vs. Ephemeral

An

*ephemeral*data structure is one for which only one version is available at a time: after an update operation, the structure as it existed before the update is lost.A

*persistent*structure is one where multiple version are simultaneously accessible: after an update, both old and new versions are available.

Functional programming is adept at implementing persistant data structures. In particular, datatypes and pattern matching make the implementation of persistent tree-like data structures remarkably straightforward.

# A Set interface

Let's think about what the interface to a persistent set should look like. We can tell that this implementation is persistent just by looking at the types of the operations.

```
class Set s where
empty :: s a
member :: Ord a => a -> s a -> Bool
insert :: Ord a => a -> s a -> s a
elements :: Ord a => s a -> [a]
```

For example, one trivial implement of sets is in terms of lists.

```
instance Set [] where
empty = undefined
member = undefined
insert = undefined
elements = undefined
```

When we define an abstract data structure like `Set`

above, we should also specify properties that *all* implementations should satisfy.

For each of these properties, we will use a `Proxy`

argument to tell quickcheck exactly which implementation it should be testing. We could use a type annotation instead (except for `prop_empty`

) but the `Proxy`

argument is a little bit easier to use.

`data Proxy (s :: * -> *) where`

For example, we can define a proxy for the list type.

```
list :: Proxy []
list = undefined
```

The empty set has no elements.

```
prop_empty :: forall s. (Set s) => Proxy s -> Bool
prop_empty _ = null (elements (empty :: s Int))
```

The elements of the set are sorted, and all of them are stored in the tree.

```
prop_elements :: (Set s) => Proxy s -> s Int -> Bool
prop_elements _ x = elements x == sort (elements x) &&
all (\y -> member y x) (elements x)
```

When we insert an element in the tree, we want to make sure that it is contained in the result.

```
prop_insert1 :: (Set s) => Proxy s -> Int -> s Int -> Bool
prop_insert1 _ x t = member x (insert x t)
```

And that the new tree also contains all of the original elements.

```
prop_insert2 :: (Set s) => Proxy s -> Int -> s Int -> Bool
prop_insert2 _ x t = all (\y -> member y t') (elements t) where
t' = insert x t
```

# Binary Search Trees

See Binary Search Trees and their implementation BST.lhs

# Balanced Trees

If our sets grow large, we may find that the simple binary tree implementation is not fast enough: in the worse case, each insert or member operation may take O(n) time!

We can do much better by keeping the trees balanced.

There are many ways of doing this. Letâ€™s look at one fairly simple (but still very fast) one that you have probably seen before in an imperative setting:

*red-black trees*.

## Red-Black Trees

See Red Black Trees and their implementation RedBlack.lhs.

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