Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs

Abstract

A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an n-vertex m-edge Helly graph can be computed with high probability in $\tilde{O}\left(m\sqrt{n}\right)$ time (i.e., subquadratic in $n+m$). In this paper, we improve that result by presenting a deterministic $O\left(m\sqrt{n}\right)$-time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity.