Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

TL;DR

This paper introduces a subquadratic worst-case update time algorithm for dynamic approximate shortest paths and related problems, improving efficiency for dense graphs and answering open questions in the field.

Contribution

It presents the first subquadratic worst-case update algorithms for dynamic approximate shortest paths, diameter, and eccentricities, using fast matrix multiplication and randomized techniques.

Findings

Achieves $ ilde O(n^{1.823}/ ext{epsilon}^2)$ worst-case update time for directed graphs.

Provides subquadratic worst-case update algorithms for eccentricities, diameter, and radius approximations.

Answers open problems for dynamic approximate shortest paths and related centrality measures.

Abstract

Consider the following distance query for an $n$-node graph $G$ undergoing edge insertions and deletions: given two sets of nodes $I$ and $J$, return the distances between every pair of nodes in $I\times J$. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient $(1+\epsilon)$-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Below are some highlights. Note that all our algorithms guarantee worst-case update time and are randomized (Monte Carlo), but do not need the oblivious adversary assumption. Subquadratic update time for SSSP, Diameter, Centralities, ect.: When we want to maintain distances from a single node explicitly (without queries), a fundamental question is to beat trivially calling Dijkstra's static algorithm after each update, taking $\Theta(n^2)$ update time on dense graphs. It was known to be improbable for exact algorithms and for combinatorial any-approximation algorithms to polynomially beat the $\Omega(n^2)$ bound (under some conjectures) [Roditty, Zwick, ESA'04; Abboud, V. Williams, FOCS'14]. Our algorithm with $I=\{s\}$ and $J=V(G)$ implies a $(1+\epsilon)$-approximation algorithm for this, guaranteeing $\tilde O(n^{1.823}/\epsilon^2)$ worst-case update time for directed graphs with positive real weights in $[1, W]$. With ideas from [Roditty, V. Williams, STOC'13], we also obtain the first subquadratic worst-case update time for $(5/3+\epsilon)$-approximating the eccentricities and $(1.5+\epsilon)$-approximating the diameter and radius for unweighted graphs (with small additive errors). [...]