Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality

Authors Victor Lecomte, Omri Weinstein



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Author Details

Victor Lecomte
  • Columbia University, New York, NY, USA
Omri Weinstein
  • Columbia University, New York, NY, USA

Acknowledgements

We want to thank the anonymous reviewers for their enthusiastic feedback and the numerous typos they spotted.

Cite AsGet BibTex

Victor Lecomte and Omri Weinstein. Settling the Relationship Between Wilber’s Bounds for Dynamic Optimality. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 68:1-68:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.68

Abstract

In FOCS 1986, Wilber proposed two combinatorial lower bounds on the operational cost of any binary search tree (BST) for a given access sequence X ∈ [n]^m. Both bounds play a central role in the ongoing pursuit of the dynamic optimality conjecture (Sleator and Tarjan, 1985), but their relationship remained unknown for more than three decades. We show that Wilber’s Funnel bound dominates his Alternation bound for all X, and give a tight Θ(lg lg n) separation for some X, answering Wilber’s conjecture and an open problem of Iacono, Demaine et. al. The main ingredient of the proof is a new symmetric characterization of Wilber’s Funnel bound, which proves that it is invariant under rotations of X. We use this characterization to provide initial indication that the Funnel bound matches the Independent Rectangle bound (Demaine et al., 2009), by proving that when the Funnel bound is constant, IRB_upRect is linear. To the best of our knowledge, our results provide the first progress on Wilber’s conjecture that the Funnel bound is dynamically optimal (1986).

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • data structures
  • binary search trees
  • dynamic optimality
  • lower bounds

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References

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