Subquadratic algorithms in minor-free digraphs: (weighted) distance oracles, decremental reachability, and more

TL;DR

This paper advances algorithms for minor-free digraphs by developing subquadratic-space distance oracles, analyzing VC set systems with pseudodimension, and creating efficient decremental reachability oracles and graph algorithms.

Contribution

It introduces the first weighted minor-free digraph distance oracle, a unified VC dimension analysis framework, and dynamic algorithms with subquadratic update times.

Findings

First exact weighted distance oracle with subquadratic space and logarithmic query time.

Unified VC dimension analysis improves understanding and bounds of set systems in digraphs.

Subquadratic algorithms for dynamic reachability, eccentricities, and Wiener index in digraphs.

Abstract

Le and Wulff-Nilsen [SODA '24] initiated a systematic study of VC set systems to unweighted $K_h$-minor-free directed graphs. We extend their results in the following ways: $\bullet$ We present the first application of VC set systems for real-weighted minor-free digraphs to build the first exact subquadratic-space distance oracle with $O(\log n)$ query time. Prior work using VC set systems only applied in unweighted and integer weighted digraphs. $\bullet$ We describe a unified system for analyzing the VC dimension of balls and the LP set system (based on Li--Parter [STOC '19]) of Le--Wulff-Nilsen [SODA '24] using pseudodimension. This is a major conceptual contribution that allows for both improving our understanding of set systems in digraphs as well as improving the bound of the LP set system in directed graphs to $h-1$. $\bullet$ We present the first application of these set systems in a dynamic setting. Specifically, we construct decremental reachability oracles with subquadratic total update time and constant query time. Prior to this work, it was not known if this was possible to construct oracles with subquadratic total update time and polylogarithmic query time, even in planar digraphs. $\bullet$ We describe subquadratic time algorithms for unweighted digraphs including (1) constructions of exact distance oracles, (2) computation of vertex eccentricities and Wiener index. The main innovation in obtaining these results is the use of dynamic string data structures.