Fully Dynamic Shortest Path Reporting Against an Adaptive Adversary

Algebraic data structures are the main subroutine for maintaining distances in fully dynamic graphs in subquadratic time. However, these dynamic algebraic algorithms generally cannot maintain the shortest paths, especially against adaptive adversaries. We present the first fully dynamic algorithm that maintains the shortest paths against an adaptive adversary in subquadratic update time. This is obtained via a combinatorial reduction that allows reconstructing the shortest paths with only a few distance estimates. Using this reduction, we obtain the following: On weighted directed graphs with real edge weights in $[1,W]$, we can maintain $(1+\epsilon)$ approximate shortest paths in $\tilde{O}(n^{1.816}\epsilon^{-2} \log W)$ update and $\tilde{O}(n^{1.741} \epsilon^{-2} \log W)$ query time. This improves upon the approximate distance data structures from [v.d.Brand, Nanongkai, FOCS'19], which only returned a distance estimate, by matching their complexity and returning an approximate shortest path. On unweighted directed graphs, we can maintain exact shortest paths in $\tilde{O}(n^{1.823})$ update and $\tilde{O}(n^{1.747})$ query time. This improves upon [Bergamaschi, Henzinger, P.Gutenberg, V.Williams, Wein, SODA'21] who could report the path only against oblivious adversaries. We improve both their update and query time while also handling adaptive adversaries. On unweighted undirected graphs, our reduction holds not just against adaptive adversaries but is also deterministic. We maintain a $(1+\epsilon)$-approximate $st$-shortest path in $O(n^{1.529} / \epsilon^2)$ time per update, and $(1+\epsilon)$-approximate single source shortest paths in $O(n^{1.764} / \epsilon^2)$ time per update. Previous deterministic results by [v.d.Brand, Nazari, Forster, FOCS'22] could only maintain distance estimates but no paths

Fast Deterministic Fully Dynamic Distance Approximation

In this paper, we develop deterministic fully dynamic algorithms for computing approximate distances in a graph with worst-case update time guarantees. In particular, we obtain improved dynamic algorithms that, given an unweighted and undirected graph $G=(V,E)$ undergoing edge insertions and deletions, and a parameter $0 < \epsilon \leq 1$, maintain $(1+\epsilon)$-approximations of the $st$-distance between a given pair of nodes $s$ and $t$, the distances from a single source to all nodes ("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the distances between all nodes ("APSP"). Our main result is a deterministic algorithm for maintaining $(1+\epsilon)$-approximate $st$-distance with worst-case update time $O(n^{1.407})$ (for the current best known bound on the matrix multiplication exponent $\omega$). This even improves upon the fastest known randomized algorithm for this problem. Similar to several other well-studied dynamic problems whose state-of-the-art worst-case update time is $O(n^{1.407})$, this matches a conditional lower bound [BNS, FOCS 2019]. We further give a deterministic algorithm for maintaining $(1+\epsilon)$-approximate single-source distances with worst-case update time $O(n^{1.529})$, which also matches a conditional lower bound. At the core, our approach is to combine algebraic distance maintenance data structures with near-additive emulator constructions. This also leads to novel dynamic algorithms for maintaining $(1+\epsilon, \beta)$-emulators that improve upon the state of the art, which might be of independent interest. Our techniques also lead to improved randomized algorithms for several problems such as exact $st$-distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st distances using new algebraic data structure