In their pioneering work, Chan, Har-Peled, and Jones (SICOMP 2020) introduced locality-sensitive ordering (LSO), and constructed an LSO with a constant number of orderings for point sets in the $d$-dimensional Euclidean space. Furthermore, their LSO could be made dynamic effortlessly under point insertions and deletions, taking $O(\log{n})$ time per update by exploiting Euclidean geometry. Their LSO provides a powerful primitive to solve a host of geometric problems in both dynamic and static settings. Filtser and Le (STOC 2022) constructed the first LSO with a constant number of orderings in the more general setting of doubling metrics. However, their algorithm is inherently static since it relies on several sophisticated constructions in intermediate steps, none of which is known to have a dynamic version. Making their LSO dynamic would recover the full generality of LSO and provide a general tool to dynamize a vast number of static constructions in doubling metrics. In this work, we give a dynamic algorithm that has $O(\log{n})$ time per update to construct an LSO in doubling metrics under point insertions and deletions. We introduce a toolkit of several new data structures: a pairwise tree cover, a net tree cover, and a leaf tracker. A key technical is stabilizing the dynamic net tree of Cole and Gottlieb (STOC 2006), a central dynamic data structure in doubling metrics. Specifically, we show that every update to the dynamic net tree can be decomposed into a few simple updates to trees in the net tree cover. As stability is the key to any dynamic algorithm, our technique could be useful for other problems in doubling metrics. We obtain several algorithmic applications from our dynamic LSO. The most notably is the first dynamic algorithm for maintaining an $k$-fault tolerant spanner in doubling metrics with optimal sparsity in optimal $O(\log{n})$ time per update.

翻译：在其开创性工作中，Chan、Har-Peled 和 Jones (SICOMP 2020) 引入了局部敏感排序（LSO）的概念，并为 $d$ 维欧几里得空间中的点集构建了具有常数数量排序的 LSO。此外，通过利用欧几里得几何性质，他们的 LSO 可以轻松实现动态化，在点插入和删除操作下每次更新仅需 $O(\log{n})$ 时间。他们的 LSO 为解决动态和静态设置下的众多几何问题提供了强大基础。Filtser 和 Le (STOC 2022) 在更一般的倍增度量设置中首次构建了具有常数数量排序的 LSO。然而，他们的算法本质上是静态的，因为它依赖于中间步骤中若干复杂的构造，而这些构造均未已知具有动态版本。将其 LSO 动态化将恢复 LSO 的完全通用性，并为倍增度量中大量静态构造的动态化提供通用工具。在本工作中，我们提出了一种动态算法，在点插入和删除操作下，每次更新仅需 $O(\log{n})$ 时间即可在倍增度量中构建 LSO。我们引入了一套包含若干新型数据结构（配对树覆盖、网树覆盖和叶节点追踪器）的工具集。一项关键技术是稳定 Cole 和 Gottlieb (STOC 2006) 提出的动态网树——这是倍增度量中的核心动态数据结构。具体而言，我们证明动态网树的每次更新均可分解为对网树覆盖中若干树的少量简单更新。由于稳定性是所有动态算法的关键，我们的技术可能对倍增度量中的其他问题具有借鉴意义。我们从动态 LSO 中获得了若干算法应用，其中最显著的是首个在倍增度量中维护 $k$ 容错生成图的动态算法，该算法以最优稀疏度实现，每次更新仅需最优的 $O(\log{n})$ 时间。