自平衡树在功能编程中最简单的是什么？

Question

I'm designing a self balancing tree in Haskell. As an exercise and because it is nice to have in your back hand.

Previously in C and Python I preferred Treaps and Splay Trees due to their simple balancing rules. I always disliked R/B Trees, since they seemed like more work than they were worth.

Now, due to the functional nature of Haskell, things seem to have changed. I can write a R/B insert function in 10 lines of code. Treaps on the other hand requires wrapping to store the random number generator, and Splay Trees are a pain to do top-down.

So I'm asking if you have experience with other types of trees? Which ones are better at utilizing the pattern matching and top-down nature of functional languages?

Answer 1

Ok, I guess there wasn't a lot of references or research for answering this question. Instead I've taken the time to try your different ideas and trees. I didn't find anything a lot better than RB trees, but perhaps that's just search bias.

The RB tree can be (insertion) balanced with four simple rules, as shown by Chris Okasaki :

balance T (T R (T R a x b) y c) z d = T R (T B a x b) y (T B c z d)
balance T (T R a x (T R b y c)) z d = T R (T B a x b) y (T B c z d)
balance T a x (T R b y (T R c z d)) = T R (T B a x b) y (T B c z d)
balance T a x (T R (T R b y c) z d) = T R (T B a x b) y (T B c z d)
balance T a x b = T B a x b

AVL trees can be balanced in a similar pattern matching way. However the rules don't compress as well:

balance T (T (T a x b   dx) y c (-1)) z d (-2) = T (T a x b dx) y (T c z d  0) 0
balance T a x (T b y (T c z d   dz)   1 )   2  = T (T a x b  0) y (T c z d dz) 0
balance T (T a x (T b y c   1 )   1 ) z d (-2) = T (T a x b -1) y (T c z d  0) 0
balance T (T a x (T b y c (-1))   1 ) z d (-2) = T (T a x b  0) y (T c z d  1) 0
balance T (T a x (T b y c   _ )   1 ) z d (-2) = T (T a x b  0) y (T c z d  0) 0
balance T a x (T (T b y c   1 ) z d (-1))   2  = T (T a x b -1) y (T c z d  0) 0
balance T a x (T (T b y c (-1)) z d (-1))   2  = T (T a x b  0) y (T c z d  1) 0
balance T a x (T (T b y c   _ ) z d (-1))   2  = T (T a x b  0) y (T c z d  0) 0
balance t = t

As AVL trees seams to generally be considered inferior to RB trees, they are probably not worth the extra hassle.

AA trees could theoretically be balanced nice and easily by:

balance T n (T n a x b) y c = T n a x (T n b y c) -- skew
balance T n a x (T n b y (T n c z d)) = T (n+1) (T n a x b) y (T n c z d) --split
balance T n a x b = T n a x b

But unfortunately Haskell don't like the overloading of n . It is possible that a less standard implementation of AA trees, not using ranks, but something more similar to R and B, would work well.

Splay trees are difficult because you need to focus on a single node, rather than the static structure of the tree. It can be done by merging insert and splay .

Treaps are also uneasy to do in a functional environment, as you don't have a global random generator, but need to keep instances in every node. This can be tackled by leaving the task of generating priorities to the client , but even then, you can't do priority comparison using pattern matching.

Answer 2

As you say Red Black trees aren't that hard to use. Have you given finger trees a look ? You might be interested in augmenting your base data structure with something like a zipper. Another tree you might find interesting is the AA tree it is a simplification of Red Black Trees.

Answer 3

It's the one that's already implemented.

There are fine implementations in Haskell of balanced trees such as Data.Map and Data.Set. Don't they fulfill your needs? Don't reimplement, reuse.

Answer 4

The OCaml standard library uses an AVL tree for its map functor. It seems as though it's easier to implement than an RB-tree if you include a remove operation.