Rendezvous of Distance-Aware Mobile Agents in Unknown Graphs
We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent’s label. It is well known that even for simple graphs of degree Δ, the rendezvous time can be exponential in Δ in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these distance-aware agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance D and the size of the smaller of the two agent labels l = min (l1, l2). Our algorithm has a time complexity of O(Δ(D + logl)) and we show an almost matching lower bound of Ω(Δ(D + logl/logΔ)) on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness.
KeywordsMobile Agent Rendezvous Synchronous Anonymous Networks Distance Oracle Lower Bounds
Unable to display preview. Download preview PDF.
- 1.Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer (2003)Google Scholar
- 6.Czyżowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 22–30 (2010)Google Scholar
- 10.Dieudonné, Y., Pelc, A.: Vincent Villain. How to meet asynchronously at polynomial cost. In: Proc 32nd Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 92–99 (2013)Google Scholar
- 13.Fraigniaud, P., Ilcinkas, D., Pelc, A.: Oracle size: A new measure of difficulty for communication problems. In: Proc. 25th Ann ACM Symposium on Principles of Distributed Computing (PODC), pp. 179–187 (2006)Google Scholar
- 16.Kranakis, E., Krizanc, D., Markou, E.: The mobile agent rendezvous problem in the ring. Morgan and Claypool Publishers (2010)Google Scholar
- 19.Stachowiak, G.: Asynchronous deterministic rendezvous on the line. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 497–508. Springer, Heidelberg (2009)Google Scholar
- 20.Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proc 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 599–608 (2007)Google Scholar