Diameter Computation on (Random) Geometric Graphs

Abstract

We present an algorithm that computes the diameter of random geometric graphs (RGGs) with expected average degree ${\Theta}(n^{\delta})$ for constant ${\delta}\in(0,1)$ in $\tilde{O}(n^{\frac{3}{2}(1+{\delta})} +n^{2 - \frac{5}{3}{\delta}})$ 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 ${\Theta}(n^{3/19})$. 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 $\tilde{O}(n^{\frac{3}{2}(1+{\delta})} + n^{2 - \frac{1}{3}{\delta}})$. 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 $\tilde{\Omega}(n^{{\delta}/3})$ over the naive $O(nm)$ algorithm, but runs in ${\Omega}(nm)$ 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.