  This paper addresses the cornerstone family of \emph{local problems} in
distributed computing, and investigates the curious gap between randomized and
deterministic solutions under bandwidth restrictions.
  Our main contribution is in providing tools for derandomizing solutions to
local problems, when the $n$ nodes can only send $O(\log n)$-bit messages in
each round of communication. We combine bounded independence, which we show to
be sufficient for some algorithms, with the method of conditional expectations
and with additional machinery, to obtain the following results.
  Our techniques give a deterministic maximal independent set (MIS) algorithm
in the CONGEST model, where the communication graph is identical to the input
graph, in $O(D\log^2 n)$ rounds, where $D$ is the diameter of the graph. The
best known running time in terms of $n$ alone is $2^{O(\sqrt{\log n})}$, which
is super-polylogarithmic, and requires large messages. For the CONGEST model,
the only known previous solution is a coloring-based $O(\Delta + \log^*
n)$-round algorithm, where $\Delta$ is the maximal degree in the graph.
  On the way to obtaining the above, we show that in the \emph{Congested
Clique} model, which allows all-to-all communication, there is a deterministic
MIS algorithm that runs in $O(\log \Delta \log n)$ rounds.%, where $\Delta$ is
the maximum degree. When $\Delta=O(n^{1/3})$, the bound improves to $O(\log
\Delta)$ and holds also for $(\Delta+1)$-coloring.
  In addition, we deterministically construct a $(2k-1)$-spanner with
$O(kn^{1+1/k}\log n)$ edges in $O(k \log n)$ rounds. For comparison, in the
more stringent CONGEST model, the best deterministic algorithm for constructing
a $(2k-1)$-spanner with $O(kn^{1+1/k})$ edges runs in $O(n^{1-1/k})$ rounds.
