We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph $G = (V, E, l)$ with polynomially bounded edge lengths, our data structure processes $|E|$ edge insertions and deletions in total time $|E|^{1 + o(1)}$ and provides query access to $|E|^{o(1)}$-approximate distances in time $\tilde{O}(1)$ per query. We produce a data structure that mimics Thorup-Zwick distance oracles [TZ'05], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We construct these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [KMPG'24] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [CKL+'24]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap our spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.

翻译：本文提出了一种用于动态近似全源最短路径（APSP）问题的简洁算法。给定边权具有多项式上界的图 $G = (V, E, l)$，我们的数据结构能够在 $|E|^{1 + o(1)}$ 的总时间内处理 $|E|$ 条边的插入与删除操作，并以每次查询 $\tilde{O}(1)$ 的时间复杂度提供 $|E|^{o(1)}$ 近似距离的查询访问。我们构建的数据结构模拟了 Thorup-Zwick 距离预言机 [TZ'05]，但具备动态性与确定性。该算法选取少量枢轴顶点，随后对于其他每个顶点，将距离计算转化为维护该顶点到其局部小邻域及最近枢轴的距离。我们通过在更小的图中表示枢轴间距离并递归处理，高效维护枢轴之间的距离。这些更小图的构建通过以下方式实现：(a) 以黑盒方式利用 Kyng-Meierhans-Probst Gutenberg [KMPG'24] 的动态保距核心图减少顶点数量；(b) 采用类似于 Chen-Kyng-Liu-Meierhans-Probst Gutenberg [CKL+'24] 的动态生成稀疏图来减少边数量。我们的动态稀疏图内部使用了 APSP 数据结构。通过在第一步选择足够大的规模缩减因子，我们能够同时自举稀疏图与动态 APSP 数据结构。值得注意的是，该方法无需借助扩展图——这一在去随机化中普遍使用的工具。