Splay trees are almost dynamically optimal

2026-07-20

AI Summary

This work investigates whether splay trees satisfy the dynamic optimality conjecture—that is, whether their performance on any access sequence is within a constant factor of that of an optimal offline dynamic binary search tree. By integrating competitive analysis of online algorithms, the potential method, and a refined recursive decomposition technique, the authors achieve the first improvement in the competitive ratio of splay trees beyond the long-standing $O(\log n)$ barrier. Specifically, they establish that splay trees are $O(\log \log n \cdot \log^2 \log \log n)$-competitive. This result breaks a forty-year theoretical impasse and provides the strongest evidence to date supporting the dynamic optimality conjecture.

Abstract

Sleator and Tarjan [JACM, 1985] conjectured that splay trees are dynamically optimal -- that on every access sequence, they perform within a constant factor of the optimal offline dynamic binary search tree. Despite four decades of work, no $o(\log n)$ competitive ratio was known. We prove that splay trees are $O(\log\log n \cdot \log^2\log\log n)=\tilde{O}(\log\log n)$-competitive.

Problem

• splay trees
• dynamic optimality
• binary search trees
• competitive ratio
• access sequence

Innovation

• splay trees
• dynamic optimality
• competitive analysis
• binary search trees
• amortized analysis