Abstract
We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimensional plane, but (with one exception) has not been investigated before for networks. Our results imply that, although these weak capabilities of robots render the problem considerably more difficult, ring exploration by a small team of robots is still possible.
We first show that, when k and n are not co-prime, the problem is not solvable in general, e.g., if k divides n there are initial placements of the robots for which gathering is impossible. We then prove that the problem is always solvable provided that n and k are co-prime, for k≥17, by giving an exploration algorithm that always terminates, starting from arbitrary initial configurations. Finally, we consider the minimum number ρ(n) of robots that can explore a ring of size n. As a consequence of our positive result we show that ρ(n) is O(logn). We additionally prove that Ω(logn) robots are necessary for infinitely many n.
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Notes
Since the ring is unoriented, agreement on only one sequence might be impossible, and the pair cannot be ordered.
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Acknowledgement
This work was done during the stay of David Ilcinkas at the Research Chair in Distributed Computing at the Université du Québec en Outaouais and at the University of Ottawa, as a postdoctoral fellow. Andrzej Pelc was partially supported by the Research Chair in Distributed Computing at the Université du Québec en Outaouais, Paola Flocchini was partially supported by the University Research Chair of the University of Ottawa. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Discovery grants.
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A preliminary version of this paper appeared in [27].
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Flocchini, P., Ilcinkas, D., Pelc, A. et al. Computing Without Communicating: Ring Exploration by Asynchronous Oblivious Robots. Algorithmica 65, 562–583 (2013). https://doi.org/10.1007/s00453-011-9611-5
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DOI: https://doi.org/10.1007/s00453-011-9611-5