  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.
