  In recent years the sleeping model came to the focus of researchers. In this
model nodes can go into a sleep state in which they spend no energy but at the
same time cannot receive or send messages, nor can they perform internal
computations. This model captures energy considerations of a problem. A problem
P is an O-LOCAL problem if, given an acyclic orientation on the edges of the
input graph, one can solve the problem as follows. Each vertex awaits the
decisions of its parents according to the given orientation and can make its
own decision in regard to P using only the information about its parents
decisions. problems and showed that for this class of problems there is a
deterministic algorithm that runs in $O(\log \Delta)$ awake time. The clock
round complexity of that algorithm is $O(\Delta^2)$.
  In this work we offer three algorithms for the bf O-LOCAL class of problems
with a trade off between awake complexity and clock round complexity. One of
these algorithms requires only $O(\Delta^{1+\epsilon})$ clock rounds for some
constant $\epsilon>0$ but still only $O(\log \Delta)$ awake time which improves
on the algorithm in \cite{BM21}. We add to this two other algorithms that trade
a higher awake complexity for lower clock round complexity. We note that the
awake time incurred is not that significant. We offer dynamic algorithms in the
sleeping model. We show three algorithms for solving dynamic problems in the
O-LOCAL class as well as an algorithm for solving any dynamic decidable
problem. We show that one can solve any {\bf O-LOCAL} problem in constant awake
time in graphs with constant neighborhood independence. Specifically, our
algorithm requires $O(K)$ awake time where $K$ is the neighborhood independence
of the input graph. Graphs with bounded neighborhood independence are well
studied with several results in recent years for several core problem in the
distributed setting.
