Splay Trees, Davenport-Schinzel Sequences, and the Deque Conjecture

TL;DR

This paper introduces a novel technique linking splay tree operations to Davenport-Schinzel sequences, providing new bounds on their performance and addressing Tarjan's deque conjecture.

Contribution

The paper develops an innovative method to analyze splay trees using sequence theory, leading to bounds on operation complexity and progress on longstanding conjectures.

Findings

Proves n deque operations take O(n alpha^*(n)) time.

Introduces a sequence-based technique for analyzing splay trees.

Suggests potential for addressing other open splay tree conjectures.

Abstract

We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a Davenport-Schinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan's deque conjecture and prove that n deque operations require O(n alpha^*(n)) time, where alpha^*(n) is the minimum number of applications of the inverse-Ackermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.