The access lemma (Sleator and Tarjan, JACM 1985) is a property of binary search trees that implies interesting consequences such as static optimality, static finger, and working set property. However, there are known corollaries of the dynamic optimality that cannot be derived via the access lemma, such as the dynamic finger, and any $o(\log n)$-competitive ratio to the optimal BST where $n$ is the number of keys. In this paper, we introduce the group access bound that can be defined with respect to a reference group access tree. Group access bounds generalize the access lemma and imply properties that are far stronger than those implied by the access lemma. For each of the following results, there is a group access tree whose group access bound Is $O(\sqrt{\log n})$-competitive to the optimal BST. Achieves the $k$-finger bound with an additive term of $O(m \log k \log \log n)$ (randomized) when the reference tree is an almost complete binary tree. Satisfies the unified bound with an additive term of $O(m \log \log n)$. Matches the unified bound with a time window $k$ with an additive term of $O(m \log k \log \log n)$ (randomized). Furthermore, we prove simulation theorem: For every group access tree, there is an online BST algorithm that is $O(1)$-competitive with its group access bound. In particular, any new group access bound will automatically imply a new BST algorithm achieving the same bound. Thereby, we obtain an improved $k$-finger bound (reference tree is an almost complete binary tree), an improved unified bound with a time window $k$, and matching the best-known bound for Unified bound in the BST model. Since any dynamically optimal BST must achieve the group access bounds, we believe our results provide a new direction towards proving $o(\log n)$-competitiveness of Splay tree and Greedy.

翻译：访问引理（Sleator与Tarjan，JACM 1985）是二叉搜索树的一个性质，它蕴含了静态最优性、静态手指和工作集性质等重要推论。然而，已知的动态最优性推论（如动态手指和任意达到最优BST的$o(\log n)$竞争比的算法，其中$n$为键数）无法通过访问引理直接导出。本文引入了群访问界的概念，该界可参照一个参考群访问树进行定义。群访问界推广了访问引理，并蕴含了远比访问引理更强的性质。对于以下每个结果，存在一个群访问树使其群访问界与最优BST达到$O(\sqrt{\log n})$竞争比：当参考树为近乎完全二叉树时，该界以$O(m \log k \log \log n)$（随机化）的附加项实现$k$-手指界；满足统一界，附加项为$O(m \log \log n)$；与带时间窗口$k$的统一界匹配，附加项为$O(m \log k \log \log n)$（随机化）。此外，我们证明了模拟定理：对于任意群访问树，存在一个在线BST算法，其与对应的群访问界达到$O(1)$竞争比。特别地，任何新的群访问界将自动蕴含一个实现相同界的新BST算法。由此，我们获得了改进的$k$-手指界（参考树为近乎完全二叉树）、带时间窗口$k$的改进统一界，并在BST模型中匹配统一界的最优已知结果。由于任何动态最优的BST必须达到群访问界，我们相信这些结果为证明伸展树与贪心算法达到$o(\log n)$竞争比提供了新方向。