# Maps and Sets from BSTs

It's easy to use a BST to implement either a map or a set ADT:

• For a map, just store a binding at each node. The nodes are ordered by the keys. The values are irrelevant to the ordering.

• For a set, just store an element at each node. The nodes are ordered by the elements.

The OCaml standard library does this for the Map and Set modules. It uses a balanced BST that is a variant of an AVL tree. AVL trees are balanced BSTs in which the height of paths is allowed to vary by at most 1. The OCaml standard library modifies that to allow the height to vary by at most 2. Like red-black trees, they achieve worst-case logarithmic performance.

Now that we have a functional map data structure, how does it compare to our imperative version, the hash table?

• Persistence: Our red-black trees are persistent, but hash tables are ephemeral.

• Performance: We get guaranteed worst-case logarithmic performance with red-black trees, but amortized, expected constant-time with hash tables. That's somewhat hard to compare given all the modifiers involved. It's also an example of a general phenomenon that persistent data structures often have to pay an extra logarithmic cost over the equivalent ephemeral data structures.

• Convenience: We have to provide an ordering function for balanced binary trees, and a hash function for hash tables. Most libraries provide a default hash function for convenience. But the performance of the hash table does depend on that hash function truly distributing keys randomly over buckets. If it doesn't, the "expected" part of the performance guarantee for hash tables is violated. So the convenience is a double-edged sword.

There isn't a clear winner here. Since the OCaml library provides both Map and Hashtbl, you get to choose.