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.

翻译：二分搜索树（BST）是最基础且广泛使用的数据结构之一。对于处理查询（搜索）序列的最优静态树，可通过动态规划计算得出。然而，当允许BST动态变化（即在搜索之间通过旋转调整结构）时，我们仍未知如何为给定序列计算最优算法（OPT）。一种候选算法被称为“贪心未来”（GF），其服务代价被猜想在（乘法）常数因子内最优。在表示BST查询的等价几何模型中，GF实际上等价于另一种称为“几何贪心”（GG）的算法。关于GF的大部分结果均通过几何模型及GG研究获得。尽管近期取得了丰硕成果，但GF竞争比的最佳下界仍为$\frac{4}{3}$。此外，有猜想认为GF与OPT的代价之间的加性差距仅与查询数量呈线性关系。本文证明了GF竞争比的下界为$2$，并证明GF与OPT代价之间的加性差距可达$\Omega(m \cdot \log\log n)$，其中$n$为树中元素数量，$m$为查询次数。