Self-Adjusting Binary Search Trees: What Makes Them Tick?

Splay trees (Sleator and Tarjan [8]) satisfy the so-called access lemma. Many of the nice properties of splay trees follow from it. What makes self-adjusting binary search trees (BSTs) satisfy the access lemma? We propose a global view of BST algorithms, which allows us to identify general structural properties of algorithms that satisfy the access lemma. As an application of our techniques, we present: (i) Unified and simpler proofs that capture all known BST algorithms satisfying the access lemma including splay trees and their generalization to the class of local algorithms (Subramanian [9], Georgakopoulos and McClurkin [6]), as well as Greedy BST, introduced by Demaine et al. [4] and shown to satisfy the access lemma by Fox [5]. (ii) A new family of algorithms based on “strict” depth-halving, shown to satisfy the access lemma. (iii) An extremely short proof for the O(logn log log n) amortized access cost for the path-balance heuristic (proposed by Sleator), matching the best known bound (Balasubramanian and Raman [2]) to a lower-order factor. Key to our results is a geometric view of how BST algorithms rearrange the search path, which reveals the underlying features that make the amortized cost logarithmic w.r.t. the sum-of-logs (SOL) potential. We obtain further structural results from this insight: for instance, we prove that locality is necessary for satisfying the access lemma, as long as the SOL potential is used. We believe that our work makes a step towards a full characterization of efficient self-adjusting BST algorithms, in particular, it shows the limitations of the SOL potential as a tool for analysis.