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A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility

Authors Jannik Castenow , Jonas Harbig , Daniel Jung , Peter Kling , Till Knollmann , Friedhelm Meyer auf der Heide

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Author Details

Jannik Castenow
  • Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Germany
Jonas Harbig
  • Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Germany
Daniel Jung
  • Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Germany
Peter Kling
  • Department of Informatics, Universität Hamburg, Germany
Till Knollmann
  • Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Germany
Friedhelm Meyer auf der Heide
  • Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Germany

Funding

This work was partially supported by the German Research Foundation (DFG) under the project number ME 872/14-1.

Cite AsGet BibTex

Jannik Castenow, Jonas Harbig, Daniel Jung, Peter Kling, Till Knollmann, and Friedhelm Meyer auf der Heide. A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.OPODIS.2022.15

Abstract

We consider a swarm of n robots in a d-dimensional Euclidean space. The robots are oblivious (no persistent memory), disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task Gathering requires that all robots reach the same, not predefined position. In the related NearGathering task, they must reach distinct positions in close proximity such that every robot sees the entire swarm. In the considered setting, Gathering can be solved in 𝒪(n + Δ²) synchronous rounds both in two and three dimensions, where Δ denotes the initial maximal distance of two robots [Hideki Ando et al., 1999; Michael Braun et al., 2020; Bastian Degener et al., 2011]. In this work, we formalize a key property of efficient Gathering protocols and use it to define λ-contracting protocols. Any such protocol gathers n robots in the d-dimensional space in 𝒪(Δ²) synchronous rounds, for d ≥ 2. For d = 1, any λ-contracting protocol gathers in optimal time 𝒪(Δ). Moreover, we prove a corresponding lower bound stating that any protocol in which robots move to target points inside the local convex hulls of their neighborhoods - λ-contracting protocols have this property - requires Ω(Δ²) rounds to gather all robots (d > 1). Among others, we prove that the d-dimensional generalization of the GTC-protocol [Hideki Ando et al., 1999] is λ-contracting. Remarkably, our improved and generalized runtime bound is independent of n and d. We also introduce an approach to make any λ-contracting protocol collision-free (robots never occupy the same position) to solve NearGathering. The resulting protocols maintain the runtime of Θ (Δ²) and work even in the semi-synchronous model. This yields the first NearGathering protocols for disoriented robots and the first proven runtime bound. In particular, combined with results from [Paola Flocchini et al., 2017] for robots with global visibility, we obtain the first protocol to solve Uniform Circle Formation (arrange the robots on the vertices of a regular n-gon) for oblivious, disoriented robots with limited visibility.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • mobile robots
  • gathering
  • limited visibility
  • runtime
  • collision avoidance
  • near-gathering

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References

  1. Sebastian Abshoff, Andreas Cord-Landwehr, Matthias Fischer, Daniel Jung, and Friedhelm Meyer auf der Heide. Gathering a closed chain of robots on a grid. In 2016 IEEE International Parallel and Distributed Processing Symposium, IPDPS 2016, Chicago, IL, USA, May 23-27, 2016, pages 689-699. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/IPDPS.2016.51.
  2. Noa Agmon and David Peleg. Fault-Tolerant Gathering Algorithms for Autonomous Mobile Robots. SIAM Journal on Computing, 36(1):56-82, January 2006. URL: https://doi.org/10.1137/050645221.
  3. Hideki Ando, Yoshinobu Oasa, Ichiro Suzuki, and Masafumi Yamashita. Distributed memoryless point convergence algorithm for mobile robots with limited visibility. IEEE Trans. Robotics Autom., 15(5):818-828, 1999. URL: https://doi.org/10.1109/70.795787.
  4. Cédric Auger, Zohir Bouzid, Pierre Courtieu, Sébastien Tixeuil, and Xavier Urbain. Certified Impossibility Results for Byzantine-Tolerant Mobile Robots. In Teruo Higashino, Yoshiaki Katayama, Toshimitsu Masuzawa, Maria Potop-Butucaru, and Masafumi Yamashita, editors, Stabilization, Safety, and Security of Distributed Systems, Lecture Notes in Computer Science, pages 178-190, Cham, 2013. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-03089-0_13.
  5. Subhash Bhagat, Sruti Gan Chaudhuri, and Krishnendu Mukhopadhyaya. Fault-tolerant gathering of asynchronous oblivious mobile robots under one-axis agreement. J. Discrete Algorithms, 36:50-62, 2016. URL: https://doi.org/10.1016/j.jda.2015.10.005.
  6. Subhash Bhagat, Sruti Gan Chaudhuri, and Krishnendu Mukhopadhyaya. Fault-Tolerant Gathering of Asynchronous Oblivious Mobile Robots under One-Axis Agreement. In M. Sohel Rahman and Etsuji Tomita, editors, WALCOM: Algorithms and Computation, Lecture Notes in Computer Science, pages 149-160, Cham, 2015. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-15612-5_14.
  7. Subhash Bhagat and Krishnendu Mukhopadyaya. Fault-tolerant Gathering of Semi-synchronous Robots. In Proceedings of the 18th International Conference on Distributed Computing and Networking, pages 1-10, Hyderabad India, January 2017. ACM. URL: https://doi.org/10.1145/3007748.3007781.
  8. Zohir Bouzid, Shantanu Das, and Sébastien Tixeuil. Gathering of Mobile Robots Tolerating Multiple Crash Faults. In 2013 IEEE 33rd International Conference on Distributed Computing Systems, pages 337-346, July 2013. URL: https://doi.org/10.1109/ICDCS.2013.27.
  9. Zohir Bouzid, Maria Gradinariu Potop-Butucaru, and Sébastien Tixeuil. Byzantine Convergence in Robot Networks: The Price of Asynchrony. In Tarek Abdelzaher, Michel Raynal, and Nicola Santoro, editors, Principles of Distributed Systems, Lecture Notes in Computer Science, pages 54-70, Berlin, Heidelberg, 2009. Springer. URL: https://doi.org/10.1007/978-3-642-10877-8_7.
  10. Zohir Bouzid, Maria Gradinariu Potop-Butucaru, and Sébastien Tixeuil. Optimal Byzantine-resilient Convergence in Unidimensional Robot Networks. Theoretical Computer Science, 411(34-36):3154-3168, July 2010. URL: https://doi.org/10.1016/j.tcs.2010.05.006.
  11. Quentin Bramas and Sébastien Tixeuil. Wait-Free Gathering Without Chirality. In Christian Scheideler, editor, Structural Information and Communication Complexity, Lecture Notes in Computer Science, pages 313-327, Cham, 2015. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-25258-2_22.
  12. Philipp Brandes, Bastian Degener, Barbara Kempkes, and Friedhelm Meyer auf der Heide. Energy-efficient strategies for building short chains of mobile robots locally. Theor. Comput. Sci., 509:97-112, 2013. URL: https://doi.org/10.1016/j.tcs.2012.10.056.
  13. Michael Braun, Jannik Castenow, and Friedhelm Meyer auf der Heide. Local gathering of mobile robots in three dimensions. In Andrea Werneck Richa and Christian Scheideler, editors, Structural Information and Communication Complexity - 27th International Colloquium, SIROCCO 2020, Paderborn, Germany, June 29 - July 1, 2020, Proceedings, volume 12156 of Lecture Notes in Computer Science, pages 63-79. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-54921-3_4.
  14. Jannik Castenow, Jonas Harbig, Daniel Jung, Peter Kling, Till Knollmann, and Friedhelm Meyer auf der Heide. A unifying approach to efficient (near)-gathering of disoriented robots with limited visibility. CoRR, abs/2206.07567, 2022. Google Scholar
  15. Jannik Castenow, Jonas Harbig, Daniel Jung, Till Knollmann, and Friedhelm Meyer auf der Heide. Gathering a euclidean closed chain of robots in linear time. In Leszek Gasieniec, Ralf Klasing, and Tomasz Radzik, editors, Algorithms for Sensor Systems - 17th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2021, Lisbon, Portugal, September 9-10, 2021, Proceedings, volume 12961 of Lecture Notes in Computer Science, pages 29-44. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-89240-1_3.
  16. Mark Cieliebak, Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. Distributed Computing by Mobile Robots: Gathering. SIAM J. Comput., 41(4):829-879, 2012. URL: https://doi.org/10.1137/100796534.
  17. Reuven Cohen and David Peleg. Convergence Properties of the Gravitational Algorithm in Asynchronous Robot Systems. SIAM J. Comput., 34(6):1516-1528, 2005. URL: https://doi.org/10.1137/S0097539704446475.
  18. Reuven Cohen and David Peleg. Convergence of Autonomous Mobile Robots with Inaccurate Sensors and Movements. SIAM J. Comput., 38(1):276-302, 2008. URL: https://doi.org/10.1137/060665257.
  19. Andreas Cord-Landwehr, Bastian Degener, Matthias Fischer, Martina Hüllmann, Barbara Kempkes, Alexander Klaas, Peter Kling, Sven Kurras, Marcus Märtens, Friedhelm Meyer auf der Heide, Christoph Raupach, Kamil Swierkot, Daniel Warner, Christoph Weddemann, and Daniel Wonisch. A New Approach for Analyzing Convergence Algorithms for Mobile Robots. In Luca Aceto, Monika Henzinger, and Jiří Sgall, editors, Automata, Languages and Programming, Lecture Notes in Computer Science, pages 650-661, Berlin, Heidelberg, 2011. Springer. URL: https://doi.org/10.1007/978-3-642-22012-8_52.
  20. Andreas Cord-Landwehr, Matthias Fischer, Daniel Jung, and Friedhelm Meyer auf der Heide. Asymptotically optimal gathering on a grid. In Christian Scheideler and Seth Gilbert, editors, Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2016, Asilomar State Beach/Pacific Grove, CA, USA, July 11-13, 2016, pages 301-312. ACM, 2016. URL: https://doi.org/10.1145/2935764.2935789.
  21. Shantanu Das, Paola Flocchini, Nicola Santoro, and Masafumi Yamashita. On the computational power of oblivious robots: forming a series of geometric patterns. In Andréa W. Richa and Rachid Guerraoui, editors, Proceedings of the 29th Annual ACM Symposium on Principles of Distributed Computing, PODC 2010, Zurich, Switzerland, July 25-28, 2010, pages 267-276. ACM, 2010. URL: https://doi.org/10.1145/1835698.1835761.
  22. Xavier Défago, Maria Gradinariu, Stéphane Messika, and Philippe Raipin-Parvédy. Fault-Tolerant and Self-stabilizing Mobile Robots Gathering. In Shlomi Dolev, editor, Distributed Computing, Lecture Notes in Computer Science, pages 46-60, Berlin, Heidelberg, 2006. Springer. URL: https://doi.org/10.1007/11864219_4.
  23. Bastian Degener, Barbara Kempkes, Peter Kling, and Friedhelm Meyer auf der Heide. Linear and competitive strategies for continuous robot formation problems. ACM Trans. Parallel Comput., 2(1):2:1-2:18, 2015. URL: https://doi.org/10.1145/2742341.
  24. Bastian Degener, Barbara Kempkes, Tobias Langner, Friedhelm Meyer auf der Heide, Peter Pietrzyk, and Roger Wattenhofer. A tight runtime bound for synchronous gathering of autonomous robots with limited visibility. In Rajmohan Rajaraman and Friedhelm Meyer auf der Heide, editors, SPAA 2011: Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Jose, CA, USA, June 4-6, 2011 (Co-located with FCRC 2011), pages 139-148. ACM, 2011. URL: https://doi.org/10.1145/1989493.1989515.
  25. Yoann Dieudonné, Franck Petit, and Vincent Villain. Leader election problem versus pattern formation problem. In Nancy A. Lynch and Alexander A. Shvartsman, editors, Distributed Computing, 24th International Symposium, DISC 2010, Cambridge, MA, USA, September 13-15, 2010. Proceedings, volume 6343 of Lecture Notes in Computer Science, pages 267-281. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15763-9_26.
  26. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors. Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7.
  27. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. Moving and computing models: Robots. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science, pages 3-14. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_1.
  28. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Giovanni Viglietta. Distributed computing by mobile robots: uniform circle formation. Distributed Comput., 30(6):413-457, 2017. URL: https://doi.org/10.1007/s00446-016-0291-x.
  29. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci., 337(1-3):147-168, 2005. URL: https://doi.org/10.1016/j.tcs.2005.01.001.
  30. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theor. Comput. Sci., 407(1-3):412-447, 2008. URL: https://doi.org/10.1016/j.tcs.2008.07.026.
  31. Taisuke Izumi, Zohir Bouzid, Sébastien Tixeuil, and Koichi Wada. The BG-simulation for Byzantine Mobile Robots, June 2011. URL: https://doi.org/10.48550/arXiv.1106.0113.
  32. Taisuke Izumi, Zohir Bouzid, Sébastien Tixeuil, and Koichi Wada. Brief Announcement: The BG-Simulation for Byzantine Mobile Robots. In David Peleg, editor, Distributed Computing, Lecture Notes in Computer Science, pages 330-331, Berlin, Heidelberg, 2011. Springer. URL: https://doi.org/10.1007/978-3-642-24100-0_32.
  33. Taisuke Izumi, Samia Souissi, Yoshiaki Katayama, Nobuhiro Inuzuka, Xavier Défago, Koichi Wada, and Masafumi Yamashita. The Gathering Problem for Two Oblivious Robots with Unreliable Compasses. SIAM J. Comput., 41(1):26-46, 2012. URL: https://doi.org/10.1137/100797916.
  34. Heinrich Jung. Ueber die kleinste kugel, die eine räumliche figur einschliesst. Journal für die reine und angewandte Mathematik, 123:241-257, 1901. URL: http://eudml.org/doc/149122.
  35. Heinrich Jung. Über den kleinsten kreis, der eine ebene figur einschließt. Journal für die reine und angewandte Mathematik, 137:310-313, 1910. URL: http://eudml.org/doc/149324.
  36. Branislav Katreniak. Convergence with Limited Visibility by Asynchronous Mobile Robots. In Adrian Kosowski and Masafumi Yamashita, editors, Structural Information and Communication Complexity - 18th International Colloquium, SIROCCO 2011, Gdansk, Poland, June 26-29, 2011. Proceedings, volume 6796 of Lecture Notes in Computer Science, pages 125-137. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22212-2_12.
  37. David G. Kirkpatrick, Irina Kostitsyna, Alfredo Navarra, Giuseppe Prencipe, and Nicola Santoro. Separating Bounded and Unbounded Asynchrony for Autonomous Robots: Point Convergence with Limited Visibility. In Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen, editors, PODC '21: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, July 26-30, 2021, pages 9-19. ACM, 2021. URL: https://doi.org/10.1145/3465084.3467910.
  38. Shouwei Li, Christine Markarian, Friedhelm Meyer auf der Heide, and Pavel Podlipyan. A continuous strategy for collisionless gathering. Theor. Comput. Sci., 852:41-60, 2021. URL: https://doi.org/10.1016/j.tcs.2020.10.037.
  39. Ji Lin, A. Stephen Morse, and Brian D. O. Anderson. The multi-agent rendezvous problem. part 1: The synchronous case. SIAM J. Control. Optim., 46(6):2096-2119, 2007. URL: https://doi.org/10.1137/040620552.
  40. Ji Lin, A. Stephen Morse, and Brian D. O. Anderson. The multi-agent rendezvous problem. part 2: The asynchronous case. SIAM J. Control. Optim., 46(6):2120-2147, 2007. URL: https://doi.org/10.1137/040620564.
  41. Linda Pagli, Giuseppe Prencipe, and Giovanni Viglietta. Getting close without touching. In Guy Even and Magnús M. Halldórsson, editors, Structural Information and Communication Complexity - 19th International Colloquium, SIROCCO 2012, Reykjavik, Iceland, June 30-July 2, 2012, Revised Selected Papers, volume 7355 of Lecture Notes in Computer Science, pages 315-326. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31104-8_27.
  42. Linda Pagli, Giuseppe Prencipe, and Giovanni Viglietta. Getting close without touching: near-gathering for autonomous mobile robots. Distributed Comput., 28(5):333-349, 2015. URL: https://doi.org/10.1007/s00446-015-0248-5.
  43. Debasish Pattanayak, Kaushik Mondal, H. Ramesh, and Partha Sarathi Mandal. Fault-Tolerant Gathering of Mobile Robots with Weak Multiplicity Detection. In Proceedings of the 18th International Conference on Distributed Computing and Networking, pages 1-4, Hyderabad India, January 2017. ACM. URL: https://doi.org/10.1145/3007748.3007786.
  44. Pavan Poudel and Gokarna Sharma. Time-optimal gathering under limited visibility with one-axis agreement. Inf., 12(11):448, 2021. URL: https://doi.org/10.3390/info12110448.
  45. Giuseppe Prencipe. Impossibility of gathering by a set of autonomous mobile robots. Theor. Comput. Sci., 384(2-3):222-231, 2007. URL: https://doi.org/10.1016/j.tcs.2007.04.023.
  46. Ichiro Suzuki and Masafumi Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. SIAM J. Comput., 28(4):1347-1363, 1999. URL: https://doi.org/10.1137/S009753979628292X.
  47. Masafumi Yamashita and Ichiro Suzuki. Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theor. Comput. Sci., 411(26-28):2433-2453, 2010. URL: https://doi.org/10.1016/j.tcs.2010.01.037.
  48. Yukiko Yamauchi, Taichi Uehara, and Masafumi Yamashita. Brief announcement: Pattern formation problem for synchronous mobile robots in the three dimensional euclidean space. In George Giakkoupis, editor, Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25-28, 2016, pages 447-449. ACM, 2016. URL: https://doi.org/10.1145/2933057.2933063.
  49. Yukiko Yamauchi and Masafumi Yamashita. Pattern Formation by Mobile Robots with Limited Visibility. In Thomas Moscibroda and Adele A. Rescigno, editors, Structural Information and Communication Complexity, Lecture Notes in Computer Science, pages 201-212, Cham, 2013. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-03578-9_17.
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