Document

# Gathering on a Circle with Limited Visibility by Anonymous Oblivious Robots

## File

LIPIcs.DISC.2020.12.pdf
• Filesize: 0.51 MB
• 17 pages

## Acknowledgements

The authors would like to thank the anonymous reviewers for greatly improving the readability of this paper.

## Cite As

Giuseppe A. Di Luna, Ryuhei Uehara, Giovanni Viglietta, and Yukiko Yamauchi. Gathering on a Circle with Limited Visibility by Anonymous Oblivious Robots. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.DISC.2020.12

## Abstract

A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant ϑ from the robot’s current location, where 0 < ϑ ≤ π (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if ϑ = π (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if ϑ ≤ π/2, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

## Subject Classification

##### ACM Subject Classification
• Computing methodologies → Distributed algorithms
• Theory of computation → Self-organization
##### Keywords
• Mobile robots
• Gathering
• limited visibility
• circle

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. N. Agmon and D. Peleg. Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM Journal on Computing, 36(1):56-82, 2006.
2. H. Ando, Y. Oasa, I. Suzuki, and M. Yamashita. Distributed memoryless point convergence algorithm for mobile robots with limited visibility. IEEE Transactions on Robotics and Automation, 15(5):818-838, 1999.
3. S. Bhagat, K. Mukhopadhyaya, and S. Mukhopadhyaya. Computation under restricted visibility. In Distributed Computing by Mobile Entities: Current Research in Moving and Computing, pages 134-183. Springer, 2019.
4. M. Cieliebak, P. Flocchini, G. Prencipe, and N. Santoro. Distributed computing by mobile robots: gathering. SIAM Journal on Computing, 41(2):829-879, 2012.
5. P. Courtieu, L. Rieg, S. Tixeuil, and X. Urbain. Impossibility of gathering, a certification. Information Processing Letters, 115(3):447-452, 2015.
6. G. D'Angelo, G. Di Stefano, and A. Navarra. Gathering on rings under the Look-Compute-Move model. Distributed Computing, 27(4):255-285, 2014.
7. G. D'Angelo, A. Navarra, and N. Nisse. A unified approach for gathering and exclusive searching on rings under weak assumptions. Distributed Computing, 30(1):17-48, 2017.
8. S. Das, G.A. Di Luna, P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Oblivious permutations on the plane. In 23rd International Conference on Principles of Distributed Systems, pages 24:1-24:16, 2019.
9. X. Défago, M. Gradinariu, S. Messika, P. Raipin-Parvédy, and S. Dolev. Fault-tolerant and self-stabilizing mobile robots gathering. In 20th International Symposium on Distributed Computing, pages 46-60, 2006.
10. B. Degener, B. Kempkes, P. Kling, F. Meyer auf der Heide, P. Pietrzyk, and R. Wattenhofer. A tight runtime bound for synchronous gathering of autonomous robots with limited visibility. In 23rd ACM Symposium on Parallelism in Algorithms and Architectures, pages 139-148, 2011.
11. G.A. Di Luna, P. Flocchini, L. Pagli, G. Prencipe, N. Santoro, and G. Viglietta. Gathering in dynamic rings. Theoretical Computer Science, 811:79-98, 2020.
12. G.A. Di Luna, P. Flocchini, N. Santoro, and G. Viglietta. TuringMobile: a turing machine of oblivious mobile robots with limited visibility and its applications. In 32nd International Symposium on Distributed Computing, pages 19:1-19:18, 2018.
13. G.A. Di Luna, P. Flocchini, N. Santoro, G. Viglietta, and M. Yamashita. Meeting in a polygon by anonymous oblivious robots. Distributed Computing (to appear), 2019.
14. G.A. Di Luna and G. Viglietta. Robots with lights. In Distributed Computing by Mobile Entities, Current Research in Moving and Computing, pages 252-277. Springer, 2019.
15. P. Flocchini. Gathering. In Distributed Computing by Mobile Entities: Current Research in Moving and Computing, pages 63-82. Springer, 2019.
16. P. Flocchini, G. Prencipe, and N. Santoro. Self-deployment of mobile sensors on a ring. Theoretical Computer Science, 402(1):67-80, 2008.
17. P. Flocchini, G. Prencipe, and N. Santoro. Distributed Computing by Oblivious Mobile Robots. Morgan & Claypool, 2012.
18. P. Flocchini, G. Prencipe, and N. Santoro, editors. Distributed Computing by Mobile Entities, Current Research in Moving and Computing. Springer, 2019.
19. P. Flocchini, G. Prencipe, N. Santoro, and G. Viglietta. Distributed computing by mobile robots: uniform circle formation. Distributed Computing, 30(6):413-457, 2017.
20. P. Flocchini, G. Prencipe, N. Santoro, and P. Widmayer. Gathering of asynchronous robots with limited visibility. Theoretical Computer Science, 337(1-3):147-168, 2005.
21. N. Fujinaga, Y. Yamauchi, S. Kijima, and M. Yamahista. Pattern formation by oblivious asynchronous mobile robots. SIAM Journal on Computing, 44(3):740-785, 2015.
22. S. Kamei, A. Lamani, F. Ooshita, S. Tixeuil, and K. Wada. Gathering on rings for myopic asynchronous robots with lights. In 23rd International Conference on Principles of Distributed Systems, pages 27:1-27:17, 2019.
23. E. Kranakis, D. Krizanc, and E. Markou. The Mobile Agent Rendezvous Problem in the Ring. Morgan and Claypool, 2010.
24. A. Monde, Y. Yamauchi, S. Kijima, and M. Yamashita. Self-stabilizing localization of the middle point of a line segment by an oblivious robot with limited visibility. In 19th International Symposium on Stabilization, Safety, and Security of Distributed Systems, pages 172-186, 2017.
25. L. Pagli, G. Prencipe, and G. Viglietta. Getting close without touching: near-gathering for autonomous mobile robots. Distributed Computing, 28(5):333-349, 2015.
26. P. Poudel and G. Sharma. Universally optimal gathering under limited visibility. In 19th International Symposium on Stabilization, Safety, and Security of Distributed Systems, pages 323-340, 2017.
27. S. Souissi, X. Défago, and M. Yamashita. Using eventually consistent compasses to gather memory-less mobile robots with limited visibility. ACM Transactions on Autonomous and Adaptive Systems, 4(1):9:1-9:27, 2009.
28. I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: formation of geometric patterns. SIAM Journal on Computing, 28(4):1347-1363, 1999.
29. Y. Yamauchi. Symmetry of anonymous robots. In Distributed Computing by Mobile Entities, Current Research in Moving and Computing, pages 109-133. Springer, 2019.
30. Y. Yamauchi, T. Uehara, S. Kijima, and M. Yamashita. Plane formation by synchronous mobile robots in the three-dimensional euclidean space. Journal of the ACM, 64(3):16:1-16:43, 2017.
31. Y. Yamauchi and M. Yamashita. Pattern formation by mobile robots with limited visibility. In 20th International Colloquium on Structural Information and Communication Complexity, pages 201-212, 2013.
X

Feedback for Dagstuhl Publishing