Self-stabilizing Localization of the Middle Point of a Line Segment by an Oblivious Robot with Limited Visibility

  • Akihiro MondeEmail author
  • Yukiko Yamauchi
  • Shuji Kijima
  • Masafumi Yamashita
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10616)


This paper poses a question about a simple localization problem, which is arisen from self-stabilizing location problems by oblivious mobile autonomous robots with limited visibility. The question is if an oblivious mobile robot on a line-segment can localize the middle point of the line-segment in finite steps observing the direction (i.e., Left or Right) and distance to the nearest end point. This problem is also akin to (a continuous version of) binary search, and could be closely related to computable real functions. Contrary to appearances, it is far from trivial if this simple problem is solvable or not, and unsettled yet. This paper is concerned with three variants of the original problem, minimally relaxing, and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.


Self-stabilization Oblivious mobile autonomous robot with limited visibility Computable real functions Continuous binary search 



This work is partly supported by JSPS KAKENHI Grant Numbers 15K15938 and 17K19982.


  1. 1.
    Ando, H., Oasa, Y., Suzuki, I., Yamashita, M.: Distributed memoryless point convergence algorithm for mobile robots. IEEE Trans. Robot. Autom. 15, 818–828 (1999)CrossRefGoogle Scholar
  2. 2.
    Ando, H., Suzuki, I., Yamashita, M.: Formation and agreement problems for synchronous mobile robots with limited visibility. In: IEEE Symposium of Intelligent Control, pp. 453–460 (1995)Google Scholar
  3. 3.
    Barriere, L., Flocchini, P., Mesa-Barrameda, E., Santoro, N.: Uniforming scattering of autonomous mobile robots in a grid. Int. J. Found. Comput. Sci. 22, 679–697 (2011)CrossRefGoogle Scholar
  4. 4.
    Cohen, R., Peleg, D.: Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput. 34, 1516–1528 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cohen, R., Peleg, D.: Local algorithms for autonomous robot systems. In: Flocchini, P., Gasieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 29–43. Springer, Heidelberg (2006). doi: 10.1007/11780823_4CrossRefGoogle Scholar
  6. 6.
    Cohen, R., Peleg, D.: Local spreading algorithms for autonomous robot systems. Theoret. Comput. Sci. 399, 71–82 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Defago, X., Konagaya, A.: Circle formation for oblivious anonymous mobile robots with no common sense of orientation. In: Proceedings of Workshop on Principles of Mobile Computing, pp. 97–104 (2002)Google Scholar
  8. 8.
    Euler, L.: Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Petropolitanae 9, 160–188 (1737)Google Scholar
  9. 9.
    Eftekhari, M., Flocchini, P., Narayanan, L., Opatrny, J., Santoro, N.: Distributed barrier coverage with relocatable sensors. In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 235–249. Springer, Cham (2014). doi: 10.1007/978-3-319-09620-9_19CrossRefGoogle Scholar
  10. 10.
    Eftekhari, M., Kranakis, E., Krizanc, D., Morales-Ponce, O., Narayanan, L., Opatrny, J., Shende, S.: Distributed algorithms for barrier coverage using relocatable sensors. Distrib. Comput. 29, 361–376 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Flocchini, P., Prencipe, G., Santoro, N.: Self-deployment algorithms for mobile sensors on a ring. Theoret. Comput. Sci. 402, 67–80 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous mobile robots with limited visibility. Theoret. Comput. Sci. 337, 147–168 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Flocchini, P., Prencipe, G., Santoro, N.: Computing by mobile robotic sensors. In: Nikoletseas, S., Rolim, J. (eds.) Theoretical Aspects of Distributed Computing in Sensor Networks. Monographs in Theoretical Computer Science. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  14. 14.
    Fujinaga, N., Yamauchi, Y., Ono, H., Kijima, S., Yamashita, M.: Pattern formation by oblivious asynchronous mobile robots. SIAM J. Comput. 44(3), 740–785 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fujisaki, G.: Field and Galois Theory. Iwanami, Tokyo (1991). (in Japanese)Google Scholar
  16. 16.
    Kleinberg, J.M.: The localization problem for mobile robots. In: Proceedings of FOCS, pp. 521–531 (1994)Google Scholar
  17. 17.
    Narkiewicz, W.: The Development of Prime Number Theory. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. 18.
    Shibata, M., Mega, T., Ooshita, F., Kakugawa, H., Masuzawa, T.: Uniform deployment of mobile agents in asynchronous rings. In: Proceedings of PODC, pp. 415–424 (2016)Google Scholar
  19. 19.
    Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots. SIAM J. Comput. 28, 1347–1363 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theoret. Comput. Sci. 411, 2433–2453 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Yamauchi, Y., Uehara, T., Kijima, S., Yamashita, M.: Plane formation by synchronous mobile robots in the three dimensional euclidean space. J. ACM 64, 16 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yamauchi, Y., Yamashita, M.: Pattern formation by mobile robots with limited visibility. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 201–212. Springer, Cham (2013). doi: 10.1007/978-3-319-03578-9_17CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Akihiro Monde
    • 1
    Email author
  • Yukiko Yamauchi
    • 1
  • Shuji Kijima
    • 1
  • Masafumi Yamashita
    • 1
  1. 1.Graduate School of Information Science and Electrical EngineeringKyushu UniversityFukuokaJapan

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