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Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

Authors Yaniv Sadeh , Haim Kaplan

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Author Details

Yaniv Sadeh
  • Tel Aviv University, Israel
Haim Kaplan
  • Tel Aviv University, Israel

Funding

The work of the authors is partially supported by Israel Science Foundation (ISF) grant number 1595-19, German Science Foundation (GIF) grant number 1367 and the Blavatnik research fund at Tel Aviv University.

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Yaniv Sadeh and Haim Kaplan. Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.53

Abstract

Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is 4/3. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of 2 on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be Ω(m ⋅ log log n) where n is the number of items in the tree and m is the number of queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Sorting and searching
Keywords
  • Binary Search Trees
  • Greedy Future
  • Geometric Greedy
  • Lower Bounds
  • Dynamic Optimality Conjecture

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