Allowing each node to communicate only once in a distributed system: shared whiteboard models
Proceedings of the twenty-fourth annual ACM symposium on Parallelism in algorithms and architectures
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In this paper we ask which properties of a distributed network can be computed from a few amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical $\cal{CONGEST}$ (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these $n$ nodes-which only knows its own ID and the IDs of its neighbors- is allowed to send a message of $O(\log n)$ bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology $G$? We show that simple questions like, "does $G$ contain a square?", "does $G$ contain a triangle?" or "Is the diameter of G at most 3?" cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of $G$ when $G$ belongs to many graph classes such as planar graphs, bounded tree width graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.