  The Hybrid network model was introduced in [Augustine et al., SODA '20] for
laying down a theoretical foundation for networks which combine two possible
modes of communication: One mode allows high-bandwidth communication with
neighboring nodes, and the other allows low-bandwidth communication over few
long-range connections at a time. This fundamentally abstracts networks such as
hybrid data centers, and class-based software-defined networks.
  Our technical contribution is a \emph{density-aware} approach that allows us
to simulate a set of \emph{oracles} for an overlay skeleton graph over a Hybrid
network.
  As applications of our oracle simulations, with additional machinery that we
provide, we derive fast algorithms for fundamental distance-related tasks. One
of our core contributions is an algorithm in the Hybrid model for computing
\emph{exact} weighted shortest paths from $\tilde O(n^{1/3})$ sources which
completes in $\tilde O(n^{1/3})$ rounds w.h.p. This improves, in both the
runtime and the number of sources, upon the algorithm of [Kuhn and Schneider,
PODC '20], which computes shortest paths from a single source in $\tilde
O(n^{2/5})$ rounds w.h.p.
  We additionally show a 2-approximation for weighted diameter and a
$(1+\epsilon)$-approximation for unweighted diameter, both in $\tilde
O(n^{1/3})$ rounds w.h.p., which is comparable to the $\tilde \Omega(n^{1/3})$
lower bound of [Kuhn and Schneider, PODC '20] for a
$(2-\epsilon)$-approximation for weighted diameter and an exact unweighted
diameter. We also provide fast distance \emph{approximations} from multiple
sources and fast approximations for eccentricities.
