  In theoretical computer science, it is a common practice to show existential
lower bounds for problems, meaning there is a family of pathological inputs on
which no algorithm can do better. However, most inputs of interest can be
solved much more efficiently, giving rise to the notion of universally optimal
algorithms, which run as fast as possible on every input. Questions on the
existence of universally optimal algorithms were first raised by Garay, Kutten,
and Peleg in FOCS '93. This research direction reemerged recently through a
series of works, including the influential work of Haeupler, Wajc, and Zuzic in
STOC '21, which resolves some of these decades-old questions in the supported
CONGEST model.
  We work in the HYBRID distributed model, which analyzes networks combining
both global and local communication. Much attention has recently been devoted
to solving distance related problems, such as All-Pairs Shortest Paths (APSP)
in HYBRID, culminating in a $\tilde \Theta(n^{1/2})$ round algorithm for exact
APSP. However, by definition, every problem in HYBRID is solvable in $D$
(diameter) rounds, showing that it is far from universally optimal.
  We show the first universally optimal algorithms in HYBRID, by presenting a
fundamental tool that solves any broadcasting problem in a universally optimal
number of rounds, deterministically. Specifically, we consider the problem in a
graph $G$ where a set of $k$ messages $M$ distributed arbitrarily across $G$,
requires every node to learn all of $M$. We show a universal lower bound and a
matching, deterministic upper bound, for any graph $G$, any value $k$, and any
distribution of $M$ across $G$.
  This broadcasting tool opens a new exciting direction of research into
showing universally optimal algorithms in HYBRID. As an example, we use it to
obtain algorithms for approximate and exact APSP in general and sparse graphs.
