Sleeping Model: Local and Dynamic Algorithms

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.