Recall: it is conjectured that splay trees are dynamically optimal (Sleator and Tarjan 1985). However, this conjecture remains unproven.

Consider a static universe of keys. Every access sequence X has an optimal BST that requires OPT(X) unit cost operations.

A BST data structure is dynamically optimal if every sequence X requires O(OPT(X)) operations.

Tango trees (Demaine et al. 2005): preferred paths of length O(log n) are represented as red-black trees in a tree of trees.

Multi-splay trees (Wang, Derryberry, and Sleator 2006): Somewhat similar to tango trees, but preferred paths are represented as splay trees instead. Better (amortized) worst-case performance.

Augmented tree (of splay trees) with Query method implemented

Only support static trees, augmentations depend on a perfect binary search tree

All modifications to the tree are made via splaying

Left2Right(node) and Right2Left(node) change the preferred child of node

Change the preferred children top-down and splay such nodes

Augmented red-black tree (of trees) with split/concatenate implemented

Challenge: the implicit tree-of-trees representation makes everything trickier!

API: lock() (read-only) and unlock() (write-only)

(Seemingly) need to implement perfect tree → tango tree conversion

After this, finishing cut/join should be relatively simple

Empirical work: benchmarks against multi-splay implementation and the like