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

TL;DR

This paper establishes new lower bounds on the performance of the Greedy-Future algorithm for dynamic binary search trees, showing it can be at least twice as costly as the optimal and the additive gap can grow significantly.

Contribution

The paper proves a lower bound of 2 on the competitive ratio of GF and demonstrates that the additive gap can be as large as (m    n), advancing understanding of GF's limitations.

Findings

Lower bound of 2 on GF's competitive ratio.

Additive gap between GF and OPT can be (m   n).

Improves previous lower bounds on GF performance.

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 $\frac{4}{3}$. 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 $\Omega(m \cdot \log\log n)$ where $n$ is the number of items in the tree and $m$ is the number of queries.