  The $\hybrid$ model was recently introduced by Augustine et al.
\cite{DBLP:conf/soda/AugustineHKSS20} in order to characterize from an
algorithmic standpoint the capabilities of networks which combine multiple
communication modes. Concretely, it is assumed that the standard $\local$ model
of distributed computing is enhanced with the feature of all-to-all
communication, but with very limited bandwidth, captured by the
node-capacitated clique ($\ncc$). In this work we provide several new insights
on the power of hybrid networks for fundamental problems in distributed
algorithms.
  First, we present a deterministic algorithm which solves any problem on a
sparse $n$-node graph in $\widetilde{\mathcal{O}}(\sqrt{n})$ rounds of
$\hybrid$. We combine this primitive with several sparsification techniques to
obtain efficient distributed algorithms for general graphs. Most notably, for
the all-pairs shortest paths problem we give deterministic $(1 + \epsilon)$-
and $\log n/\log \log n$-approximate algorithms for unweighted and weighted
graphs respectively with round complexity $\widetilde{\mathcal{O}}(\sqrt{n})$
in $\hybrid$, closely matching the performance of the state of the art
randomized algorithm of Kuhn and Schneider \cite{10.1145/3382734.3405719}.
Moreover, we make a connection with the Ghaffari-Haeupler framework of
low-congestion shortcuts \cite{DBLP:conf/soda/GhaffariH16}, leading -- among
others -- to a $(1 + \epsilon)$-approximate algorithm for Min-Cut after
$\log^{\mathcal{O}(1)}n$ rounds, with high probability, even if we restrict
local edges to transfer $\mathcal{O}(\log n)$-bits per round. Finally, we prove
via a reduction from the set disjointness problem that
$\widetilde{\Omega}(n^{1/3})$ rounds are required to determine the radius of an
unweighted graph, as well as a $(3/2 - \epsilon)$-approximation for weighted
graphs.
