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Can a Skywalker Localize the Midpoint of a Rope?

Published:18 July 2021Publication History
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Abstract

This article poses a question about a simple localization problem. The question is if an oblivious walker on a line segment can localize the midpoint of the line segment in a finite number of steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem arises from self-stabilizing location problems by autonomous mobile robots with limited visibility, which is an abstract model attracting a wide interest in distributed computing. Contrary to appearances, it is far from trivial whether this simple problem is solvable, and it is not settled yet. This article is concerned with three variants of the problem with a minimal relaxation and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.

References

  1. Hideki Ando, Yoshinobu Oasa, Ichiro Suzuki, and Masafumi Yamashita. 1999. Distributed memoryless point convergence algorithm for mobile robots. IEEE Trans. Robot. Autom. 15, 5, (Oct. 1999), 818–828. DOI:DOI:https://dx.doi.org/10.1109/70.795787Google ScholarGoogle ScholarCross RefCross Ref
  2. Hideki Ando, Ichiro Suzuki, and Masafumi Yamashita. 1995. Formation and agreement problems for synchronous mobile robots with limited visibility. In Proceedings of the 10th IEEE Symposium of Intelligent Control (ISIC’95).453–460. DOI:DOI:https://dx.doi.org/10.1109/ISIC.1995.525098Google ScholarGoogle ScholarCross RefCross Ref
  3. Lali Barriere, Paola Flocchini, Eduardo Mesa-Barrameda, and Nicola Santoro. 2011. Uniforming scattering of autonomous mobile robots in a grid. Int. J. Found. Comput. Sci. 22, 3, (2011), 679–697. DOI:DOI:https://doi.org/10.1142/S0129054111008295Google ScholarGoogle ScholarCross RefCross Ref
  4. Reuven Cohen and David Peleg. 2005. Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput. 34, 6, (2005), 1516–1528. DOI:DOI:https://doi.org/10.1137/S0097539704446475Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Reuven Cohen and David Peleg. 2006. Local algorithms for autonomous robot systems. In Proceedings of the 13th International Colloquium on Structural Information and Communication Complexity (SIROCCO’06).29–43. DOI:DOI:https://doi.org/10.1007/11780823_4Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Reuven Cohen and David Peleg. 2008. Local spreading algorithms for autonomous robot systems. Theor. Comput. Sci. 399, 1–2, (June 2008), 71–82. DOI:DOI:https://doi.org/10.1016/j.tcs.2008.02.007Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Xavier Défago and Akihiko Konagaya. 2002. Circle formation for oblivious anonymous mobile robots with no common sense of orientation. In Proceedings of the ACM Workshop on Principles of Mobile Computing (POMC’02).97–104. DOI:DOI:https://doi.org/10.1145/584490.584509Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Edsger W. Dijkstra. 1974. Self-stabilizing systems in spite of distributed control. Commun. ACM 17, 11 (Nov. 1974), 643–644. DOI:https://doi.org/10.1145/361179.361202Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Leonhard Euler. 1737. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9, (1744), 160–188. DOI:https://doi.org/10.1145/361179.361202Google ScholarGoogle Scholar
  10. Mohsen Eftekhari, Paola Flocchini, Lata Narayanan, Jaroslav Opatrny, and Nicola Santoro. 2014. Distributed barrier coverage with relocatable sensors. In Proceedings of the 21st International Colloquium on Structural Information and Communication Complexity (SIROCCO’14).235–249. DOI:DOI:https://doi.org/10.1007/978-3-319-09620-9_19Google ScholarGoogle ScholarCross RefCross Ref
  11. Mohsen Eftekhari, Evangelos Kranakis, Danny Krizanc, Oscar Morales-Ponce, Lata Narayanan, Jaroslav Opatrny, and Sunil Shende. 2016. Distributed algorithms for barrier coverage using relocatable sensors. Distrib. Comput. 29, (Apr. 2016), 361–376. DOI:DOI:https://doi.org/10.1007/s00446-016-0267-xGoogle ScholarGoogle ScholarDigital LibraryDigital Library
  12. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. 2008. Self-deployment algorithms for mobile sensors on a ring. Theor. Comput. Sci. 402, 1, (July 2008), 67–80. DOI:DOI:https://doi.org/10.1016/j.tcs.2008.03.006Google ScholarGoogle Scholar
  13. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. 2005. Gathering of asynchronous mobile robots with limited visibility. Theor. Comput. Sci. 337, 1-3, (June 2005), 147–168. DOI:DOI:https://doi.org/10.1016/j.tcs.2005.01.001Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. 2011. Computing by Mobile Robotic Sensors.Springer, Berlin.Google ScholarGoogle Scholar
  15. Nao Fujinaga, Yukiko Yamauchi, Hirotaka Ono, Shuji Kijima, and Masafumi Yamashita. 2015. Pattern formation by oblivious asynchronous mobile robots. SIAM J. Comput. 44, 3, (2015). 740–785. DOI:DOI:https://doi.org/10.1137/140958682Google ScholarGoogle ScholarCross RefCross Ref
  16. Jon Michael Kleinberg. 1994. The localization problem for mobile robots. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS’94).521–531. DOI:DOI:https://doi.org/10.1109/SFCS.1994.365739Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Akihiro Monde, Yukiko Yamauchi, Shuji Kijima, and Masafumi Yamashita. 2017. Self-stabilizing localization of the middle point of a line segment by an oblivious robot with limited visibility. In Proceedings of the 19th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS’17).172–186. DOI:DOI:https://doi.org/10.1007/978-3-319-69084-1_12Google ScholarGoogle ScholarCross RefCross Ref
  18. Władysław Narkiewicz. 2000. The Development of Prime Number Theory.Springer-Verlag, Berlin.Google ScholarGoogle ScholarCross RefCross Ref
  19. Masahiro Shibata, Toshiya Mega, Fukuhito Ooshita, Hirotsugu Kakugawa, and Toshimitsu Masuzawa. 2018. Uniform deployment of mobile agents in asynchronous rings. J. Parallel Distrib. Comput. 119, (Sept. 2018), 92–106. DOI:DOI:https://doi.org/10.1016/j.jpdc.2018.03.008Google ScholarGoogle ScholarCross RefCross Ref
  20. Ichiro Suzuki and Masafumi Yamashita. 1999. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28, 4, (1999), 1347–1363. DOI:DOI:https://doi.org/10.1137/S009753979628292XGoogle ScholarGoogle ScholarDigital LibraryDigital Library
  21. Masafumi Yamashita and Ichiro Suzuki. 2010. Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theor. Comput. Sci. 411, 26–28, (2010), 2433–2453. DOI:DOI:https://doi.org/10.1016/j.tcs.2010.01.037Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Yukiko Yamauchi, Taichi Uehara, Shuji Kijima, and Masafumi Yamashita. 2017. Plane formation by synchronous mobile robots in the three dimensional Euclidean space. J. ACM 64, 3 (June 2017). DOI:DOI:https://doi.org/10.1145/3060272Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Yukiko Yamauchi and Masafumi Yamashita. 2013. Pattern formation by mobile robots with limited visibility. In Proceedings of the 20th International Colloquium on Structural Information and Communication Complexity (SIROCCO’13).201–212. DOI:DOI:https://doi.org/10.1007/978-3-319-03578-9_17Google ScholarGoogle ScholarCross RefCross Ref

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