We present a general toolbox, based on new vertex sparsifiers, for designing data structures to maintain shortest paths in dynamic graphs. In an $m$-edge graph undergoing edge insertions and deletions, our data structures give the first algorithms for maintaining (a) $m^{o(1)}$-approximate all-pairs shortest paths (APSP) with \emph{worst-case} update time $m^{o(1)}$ and query time $\tilde{O}(1)$, and (b) a tree $T$ that has diameter no larger than a subpolynomial factor times the diameter of the underlying graph, where each update is handled in amortized subpolynomial time. In graphs undergoing only edge deletions, we develop a simpler and more efficient data structure to maintain a $(1+\epsilon)$-approximate single-source shortest paths (SSSP) tree $T$ in a graph undergoing edge deletions in amortized time $m^{o(1)}$ per update. Our data structures are deterministic. The trees we can maintain are not subgraphs of $G$, but embed with small edge congestion into $G$. This is in stark contrast to previous approaches and is useful for algorithms that internally use trees to route flow. To illustrate the power of our new toolbox, we show that our SSSP data structure gives simple deterministic implementations of flow-routing MWU methods in several contexts, where previously only randomized methods had been known. To obtain our toolbox, we give the first algorithm that, given a graph $G$ undergoing edge insertions and deletions and a dynamic terminal set $A$, maintains a vertex sparsifier $H$ that approximately preserves distances between terminals in $A$, consists of at most $|A|m^{o(1)}$ vertices and edges, and can be updated in worst-case time $m^{o(1)}$. Crucially, our vertex sparsifier construction allows us to maintain a low edge-congestion embedding of $H$ into $G$, which is needed for our applications.

翻译：我们提出一个基于新型顶点稀疏化的通用工具箱，用于设计动态图中维护最短路径的数据结构。在经历边插入和删除的$m$边图中，我们的数据结构首次实现了以下算法：(a) 具有*最坏情况*更新时间为$m^{o(1)}$、查询时间为$\tilde{O}(1)$的$m^{o(1)}$-近似全源最短路径（APSP），以及(b) 一棵直径不超过底层图直径次多项式因子的树$T$，每次更新在均摊次多项式时间内完成。在仅经历边删除的图中，我们设计了一种更简单高效的数据结构，可在每次更新均摊时间$m^{o(1)}$内维护一棵$(1+\epsilon)$-近似单源最短路径（SSSP）树$T$。我们的数据结构是确定性的。所维护的树并非$G$的子图，而是以低边拥塞嵌入到$G$中。这与先前方法形成鲜明对比，且对内部使用树进行流路由的算法尤为实用。为展示新工具箱的能力，我们证明SSSP数据结构可在多种场景下为流路由MWU方法提供简单确定性实现——此前仅已知随机化方法。为构建该工具箱，我们提出首个算法：给定经历边插入和删除的图$G$与动态终端集$A$，可维护顶点稀疏化$H$，其近似保留$A$中终端间距离、包含至多$|A|m^{o(1)}$个顶点和边，且可在最坏情况时间$m^{o(1)}$内更新。关键的是，我们的顶点稀疏化构造支持将$H$以低边拥塞嵌入$G$，这对其应用至关重要。