  Distributed graph algorithms that separately optimize for either the number
of rounds used or the total number of messages sent have been studied
extensively. However, algorithms simultaneously efficient with respect to both
measures have been elusive. For example, only very recently was it shown that
for Minimum Spanning Tree (MST), an optimal message and round complexity is
achievable (up to polylog terms) by a single algorithm in the CONGEST model of
communication.
  In this paper we provide algorithms that are simultaneously round- and
message-optimal for a number of well-studied distributed optimization problems.
Our main result is such a distributed algorithm for the fundamental primitive
of computing simple functions over each part of a graph partition. From this
algorithm we derive round- and message-optimal algorithms for multiple
problems, including MST, Approximate Min-Cut and Approximate Single Source
Shortest Paths, among others. On general graphs all of our algorithms achieve
worst-case optimal $\tilde{O}(D+\sqrt n)$ round complexity and $\tilde{O}(m)$
message complexity. Furthermore, our algorithms require an optimal
$\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages on planar, genus-bounded,
treewidth-bounded and pathwidth-bounded graphs.
