Allowing Multiple Rounds in the Shared Whiteboard Model: Some More Impossibility Results

The shared whiteboard model for distributed computing is one of the recent interesting models to be proposed (See Becker et al. (SPAA 2012)). In its basic form, this model allows all nodes to write a message of no more than O(log n) bits on a whiteboard that every node can read. However, each node can write at most once. In this model, a variety of problems from graphs are shown to be either possible or impossible. In this paper we extend the work of Becker et al. to allow for nodes to write on the shared whiteboard more than once. However, each node can write at most O(log n) bits at any one instant. Interestingly, in this model, we show that allowing just two rounds of writing on the whiteboard, one can color the vertices of a d-degenerate graph using d+1 colors. Similarly, we show that two rounds suffice to find maximal independent set (MIS), whereas 2-ruling sets can be computed in one round in simultaneous synchronous models. Finally, we show that for finding connected components in a graph, even O(1) rounds is not enough in general. We show that any deterministic algorithm that follows certain rules requires at least Omega(log nlog log n) rounds to find the connected components of an n-vertex graph. At the same time, we show the existence of a O(log n) round algorithm for the same. Thus, our results indicate that the multi-round shared whiteboard model has interesting consequences.