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

TL;DR

This paper presents deterministic and parameterized linear-time algorithms for computing all eccentricities, radius, and diameter in Helly graphs, significantly improving efficiency over previous probabilistic methods.

Contribution

It introduces a deterministic ${ m O}(m oot n)$ time algorithm and a parameterized linear-time algorithm based on hyperbolicity for all eccentricity computations in Helly graphs.

Findings

Deterministic ${ m O}(m oot n)$ time algorithm for all eccentricities.

Linear-time algorithm parameterized by hyperbolicity $oldsymbol{igO(oldsymbol{ ext{delta}} m)}$.

Structural properties of Helly graphs enable these efficient algorithms.

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. It is also known that every graph isometrically embeds into a Helly graph, making the latter an important class of graphs in Metric Graph Theory. 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 $G$ can be computed with high probability in $\tilde{\mathcal O}(m\sqrt{n})$ time (i.e., subquadratic in $n+m$). In this paper, we improve that result by presenting a deterministic ${\mathcal O}(m\sqrt{n})$ 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 $\delta$. More specifically, we show that the radius and a central vertex of an $m$-edge $\delta$-hyperbolic Helly graph $G$ can be computed in $\mathcal O(\delta m)$ time and that all vertex eccentricities in $G$ can be computed in $\mathcal O(\delta^2 m)$ time. To show this more general result, we heavily use our new structural properties obtained for Helly graphs.