Diameter Computation on (Random) Geometric Graphs

2026-03-17

AI Summary

This work addresses the long-standing absence of subquadratic-time algorithms for computing the diameter of random geometric graphs (RGGs) in sparse to moderately dense regimes. The authors propose a general framework based on balanced separators with favorable metric properties, yielding the first subquadratic-time diameter algorithm for RGGs and establishing a deterministic analytical framework for graph properties in geometric settings. By integrating combinatorial structures, the RGG model, and diameter algorithms such as iFUB, they reduce the diameter computation complexity to Õ(n^{33/19}) ≈ Õ(n^{1.737}) for RGGs with average degree Θ(n^{3/19}), improving upon the O*(n^{1.944}) bound known for unit disk graphs. Moreover, they provide the first theoretical guarantee that iFUB achieves an expected speedup of Õ(n^{δ/3}) on geometric graphs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present an algorithm that computes the diameter of random geometric graphs (RGGs) with expected average degree $Θ(n^δ)$ for constant $δ\in(0,1)$ in $\tilde{O}(n^{\frac{3}{2}(1+δ)} +n^{2 - \frac{5}{3}δ})$ time, asymptotically almost surely. This brings the running time down to $\tilde{O}(n^{\frac{33}{19}})\approx \tilde{O}(n^{1.737})$ for average degree $Θ(n^{3/19})$. To the best of our knowledge, this constitutes the first such bound for RGGs and for a substantial range of average degrees, it is notably smaller than the recent bound of $O^*(n^{2-1/18}) \approx O^*(n^{1.944})$ by Chan et al. (FOCS 2025) for the more general class of all unit disk graphs. Our algorithm also works on RGGs with the flat torus as ground space, with a running time in $\tilde{O}(n^{\frac{3}{2}(1+δ)} + n^{2 - \frac{1}{3}δ})$. While our bounds on random geometric graphs are interesting in their own right, they are only an application of our main contribution: A general framework of deterministic graph properties that enable efficient diameter computation. Our properties are based on the existence of balanced separators that are well-behaved regarding the metric space defined by the graph and can be seen as a distillation of the combinatorial features a graph gets from having an underlying geometry. As a by-product of verifying that RGGs fit into our framework, we also derive running time bounds for iFUB, a diameter algorithm by Crescenzi et al. (TCS 2013) that is highly efficient on real-world graphs. We show that a.a.s.\ iFUB achieves a speedup in $\tildeΩ(n^{δ/3})$ over the naive $O(nm)$ algorithm, but runs in $Ω(nm)$ time on torus RGGs. This constitutes the first theoretical analysis in a geometric setting and confirms prior empirical evidence, thus suggesting geometry as a reasonable model for certain real-world inputs.

Problem

• diameter computation
• random geometric graphs
• graph algorithms
• unit disk graphs
• geometric graph properties

Innovation

• random geometric graphs
• diameter computation
• balanced separators
• metric structure
• iFUB algorithm