  Diameter, radius and eccentricities are fundamental graph parameters, which
are extensively studied in various computational settings. Typically, computing
approximate answers can be much more efficient compared with computing exact
solutions. In this paper, we give a near complete characterization of the
trade-offs between approximation ratios and round complexity of distributed
algorithms for approximating these parameters, with a focus on the weighted and
directed variants.
  Furthermore, we study \emph{bi-chromatic} variants of these parameters
defined on a graph whose vertices are colored either red or blue, and one
focuses only on distances for pairs of vertices that are colored differently.
Motivated by applications in computational geometry, bi-chromatic diameter,
radius and eccentricities have been recently studied in the sequential setting
[Backurs et al. STOC'18, Dalirrooyfard et al. ICALP'19]. We provide the first
distributed upper and lower bounds for such problems.
  Our technical contributions include introducing the notion of
\emph{approximate pseudo-center}, which extends the \emph{pseudo-centers} of
[Choudhary and Gold SODA'20], and presenting an efficient distributed algorithm
for computing approximate pseudo-centers. On the lower bound side, our
constructions introduce the usage of new functions into the framework of
reductions from 2-party communication complexity to distributed algorithms.
