A Nearly Linear Time Construction of Approximate Single-Source Distance Sensitivity Oracles
Kaito Harada, Naoki Kitamura, Taisuke Izumi, Toshimitsu Masuzawa
TL;DR
The paper addresses single-source replacement path under vertex faults by introducing a deterministic construction of a (1+ε)-approximate VSDO for directed weighted graphs with polynomially bounded weights, achieving near-linear construction time and polylogarithmic query time. The core method combines centroid-based divide-and-conquer with progressive Dijkstra, building three interdependent sub-oracles (G1, G2, and a first-case DP-like component) in a recursive framework that preserves correctness while avoiding explicit SSRP computation. The result yields an oracle of size O(ε^{-1} n log^3 n log(nW)) with construction time O(ε^{-1} log^4 n log(nW) (m + n ε^{-1} log^3 n log(nW))) and query time O(log^2 n log(ε^{-1} log(nW))), enabling efficient (1+ε)-approximate SSRP in near-linear time and breaking previous mn-based lower bounds under approximation. This advances fault-tolerant path computation with practical implications for dynamic, reliability-critical networks and large-scale graph analysis under uncertainty.
Abstract
An \emph{$α$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$α$-VSDO}) for a weighted input graph $G=(V, E, w)$ and a source vertex $s \in V$ is the data structure answering an $α$-approximate distance from $s$ to $t$ in $G-x$ for any given query $(x, t) \in V \times V$. It is a data structure version of the so-called single-source replacement path problem (SSRP). In this paper, we present a new \emph{nearly linear-time} algorithm of constructing a $(1 + ε)$-VSDO for any directed input graph with polynomially bounded integer edge weights. More precisely, the presented oracle attains $\tilde{O}(m \log (nW)/ ε+ n \log^2 (nW)/ε^2)$ construction time, $\tilde{O}(n \log (nW) / ε)$ size, and $\tilde{O}(1/ε)$ query time, where $n$ is the number of vertices, $m$ is the number of edges, and $W$ is the maximum edge weight. These bounds are all optimal up to polylogarithmic factors. To the best of our knowledge, this is the first non-trivial algorithm for SSRP/VSDO beating $\tilde{O}(mn)$ computation time for directed graphs with general edge weight functions, and also the first nearly linear-time construction breaking approximation factor 3. Such a construction has been unknown even for undirected and unweighted graphs. In addition, our result implies that the known conditional lower bounds for the exact SSRP computation does not apply to the case of approximation.