Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
- 1.
We conjecture that Problem 1 is unsolvable. To avoid ambiguity, especially for an impossibility proof (in the future), we give there a formal description of the problem.
- 2.
Here, \(x \mapsto f(\phi (D,x))\) represents a transition of the robot on the interval \([-D,D]\).
References
Ando, H., Oasa, Y., Suzuki, I., Yamashita, M.: Distributed memoryless point convergence algorithm for mobile robots. IEEE Trans. Robot. Autom. 15, 818–828 (1999)
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)
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)
Cohen, R., Peleg, D.: Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput. 34, 1516–1528 (2005)
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_4
Cohen, R., Peleg, D.: Local spreading algorithms for autonomous robot systems. Theoret. Comput. Sci. 399, 71–82 (2008)
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)
Euler, L.: Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Petropolitanae 9, 160–188 (1737)
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_19
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)
Flocchini, P., Prencipe, G., Santoro, N.: Self-deployment algorithms for mobile sensors on a ring. Theoret. Comput. Sci. 402, 67–80 (2008)
Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous mobile robots with limited visibility. Theoret. Comput. Sci. 337, 147–168 (2005)
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)
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)
Fujisaki, G.: Field and Galois Theory. Iwanami, Tokyo (1991). (in Japanese)
Kleinberg, J.M.: The localization problem for mobile robots. In: Proceedings of FOCS, pp. 521–531 (1994)
Narkiewicz, W.: The Development of Prime Number Theory. Springer, Heidelberg (2000)
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)
Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots. SIAM J. Comput. 28, 1347–1363 (1999)
Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by oblivious anonymous mobile robots. Theoret. Comput. Sci. 411, 2433–2453 (2010)
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)
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_17
Acknowledgement
This work is partly supported by JSPS KAKENHI Grant Numbers 15K15938 and 17K19982.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Monde, A., Yamauchi, Y., Kijima, S., Yamashita, M. (2017). Self-stabilizing Localization of the Middle Point of a Line Segment by an Oblivious Robot with Limited Visibility. In: Spirakis, P., Tsigas, P. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2017. Lecture Notes in Computer Science(), vol 10616. Springer, Cham. https://doi.org/10.1007/978-3-319-69084-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-69084-1_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69083-4
Online ISBN: 978-3-319-69084-1
eBook Packages: Computer ScienceComputer Science (R0)