  Algorithms with advice have received ample attention in the distributed and
online settings, and they have recently proven useful also in dynamic settings.
In this work we study local computation with advice: the goal is to solve a
graph problem $\Pi$ with a distributed algorithm in $f(\Delta)$ communication
rounds, for some function $f$ that only depends on the maximum degree $\Delta$
of the graph, and the key question is how many bits of advice per node are
needed. Our main results are:
  - Any locally checkable labeling problem can be solved in graphs with
sub-exponential growth with only $1$ bit of advice per node. Moreover, we can
make the set of nodes that carry advice bits arbitrarily sparse, that is, we
can make arbitrarily small the ratio between nodes carrying a 1 and the nodes
carrying a 0. - The assumption of sub-exponential growth is necessary: assuming
the Exponential-Time Hypothesis, there are LCLs that cannot be solved in
general with any constant number of bits per node. - In any graph we can find
an almost-balanced orientation (indegrees and outdegrees differ by at most one)
with $1$ bit of advice per node, and again we can make the advice arbitrarily
sparse. - As a corollary, we can also compress an arbitrary subset of edges so
that a node of degree $d$ stores only $d/2 + 2$ bits, and we can decompress it
locally, in $f(\Delta)$ rounds. - In any graph of maximum degree $\Delta$, we
can find a $\Delta$-coloring (if it exists) with $1$ bit of advice per node,
and again, we can make the advice arbitrarily sparse. - In any $3$-colorable
graph, we can find a $3$-coloring with $1$ bit of advice per node. Here, it
remains open whether we can make the advice arbitrarily sparse.
  Our work shows that for many problems the key threshold is not whether we can
achieve, say, $1$ bit of advice per node, but whether we can make the advice
arbitrarily sparse.
