Time of Anonymous Rendezvous in Trees: Determinism vs. Randomization
Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and move along its edges with the goal of meeting at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We study optimal time of completing this rendezvous task. For deterministic rendezvous we seek algorithms that achieve rendezvous whenever possible, while for randomized rendezvous we seek almost safe algorithms, which achieve rendezvous with probability at least 1 − 1/n in n-node trees, for sufficiently large n.
We construct a deterministic algorithm that achieves rendezvous in time O(n) in n-node trees, whenever rendezvous is feasible, and we show that this time cannot be improved in general, even when agents start at distance 1 in bounded degree trees. We also show an almost safe algorithm that achieves rendezvous in time O(n) for arbitrary starting positions in any n-node tree. We then analyze when randomization can help to speed up rendezvous. For n-node trees of known constant maximum degree and for a known constant upper bound on the initial distance between the agents, we show an almost safe algorithm achieving rendezvous in time O(logn). By contrast, we show that for some trees, every almost safe algorithm must use time Ω(n), even for initial distance 1. This shows an exponential gap between randomized rendezvous time in trees of bounded degree and in arbitrary trees. Such a gap does not occur for deterministic rendezvous.
All our upper bounds hold when agents start with an arbitrary delay, controlled by the adversary, and all our lower bounds hold even when agents start simultaneously.
KeywordsInitial Position Mobile Agent Central Node Deterministic Algorithm Port Number
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- 1.Alpern, S., Gal, S.: The theory of search games and rendezvous. Int. Series in Operations research and Management Science. Kluwer Academic Publisher (2002)Google Scholar
- 7.Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: Log-space rendezvous in arbitrary graphs. In: Proc. 29th Annual ACM Symposium on Principles of Distributed Computing (PODC 2010), pp. 450–459 (2010)Google Scholar
- 8.Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: Proc. 21st Ann. ACM Symp. on Discr. Algorithms (SODA 2010), pp. 22–30 (2010)Google Scholar
- 9.Degener, B., Kempkes, B., Meyer auf der Heide, F.: A local O(n2) gathering algorithm. In: Proc. 22nd Ann. ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2010), pp. 217–223 (2010)Google Scholar
- 13.Fraigniaud, P., Pelc, A.: Delays induce an exponential memory gap for rendezvous in trees. ArXiv: 1102.0467v1 (2011)Google Scholar
- 16.Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Mobile agent rendezvous in a ring. In: Proc. 23rd Int. Conf. on Distr. Computing Systems (ICDCS 2003), pp. 592–599 (2003)Google Scholar
- 17.Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proc. 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 599–608 (2007)Google Scholar
- 18.Yao, A.C.-C.: Probabilistic computations: Towards a unified measure of complexity. In: Proc. 18th Annual IEEE Conference on Foundations of Computer Science (FOCS 1977), pp. 222–227 (1977)Google Scholar