The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in $\mathbb{R}^2$, which are topological summaries obtained by computing persistent homology from every direction in $\mathbb{S}^1$.

翻译：动力学数据结构（KDS）框架是维护连续运动物体各种几何构型的强大工具。本文提出了一种新型KDS实现——动力学沙漏，专门用于计算几何匹配问题中的瓶颈距离。我们详细阐述了处理一般图所需的事件与更新机制，并辅以复杂度分析。进一步地，通过将动力学沙漏应用于计算$\mathbb{R}^2$中两个持续同调变换（PHT）之间的瓶颈距离，我们展示了其实用性。这些PHT是拓扑摘要，通过从$\mathbb{S}^1$中所有方向计算持续同调获得。