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Parallel and Distributed Processing Symposium, International (2011)
Anchorage, Alaska USA
May 16, 2011 to May 20, 2011
ISSN: 1530-2075
ISBN: 978-0-7695-4385-7
pp: 508-514
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.

I. Todinca, I. Rapaport, K. Suchan, N. Nisse, F. Becker and M. Matamala, "Adding a Referee to an Interconnection Network: What Can(not) Be Computed in One Round," 25th IEEE International Parallel & Distributed Processing Symposium (IPDPS 2011)(IPDPS), Anchorage, AK, 2011, pp. 508-514.
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