Splay trees are almost dynamically optimal

Petr Chmel; Bernhard Haeupler; Richard Hladík; Michal Koucký; Antti Roeyskoe; Václav Rozhoň; Ondřej Sladký; Robert E. Tarjan

Splay trees are almost dynamically optimal

Data Structures and Algorithms 2026-07-20 v1

Authors: Petr Chmel , Bernhard Haeupler , Richard Hladík , Michal Koucký , Antti Roeyskoe , Václav Rozhoň , Ondřej Sladký , Robert E. Tarjan

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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.

Cite

@article{arxiv.2607.18498,
  title  = {Splay trees are almost dynamically optimal},
  author = {Petr Chmel and Bernhard Haeupler and Richard Hladík and Michal Koucký and Antti Roeyskoe and Václav Rozhoň and Ondřej Sladký and Robert E. Tarjan},
  journal= {arXiv preprint arXiv:2607.18498},
  year   = {2026}
}