Skip to main content
SpringerLink
Log in

Better Upper Bounds for Searching on a Line with Byzantine Robots

  • Chapter
  • First Online:
Complexity and Approximation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12000))

Abstract

Searching on a line with Byzantine robots was first posed by Czyzowicz et al. in [13]: Suppose there are n robots searching on an infinite line to find a target which is unknown to the robots. At the beginning all robots stay at the origin and then they can start to search with maximum speed 1. Unfortunately, f of them are Byzantine fault, which means that they may ignore the target when passing it or lie that they find the target. Therefore, the target is found if at least \(f+1\) robots claim that they find the target at the same location. The aim is to design a parallel algorithm to minimize the competitive ratio S(n, f), the ratio between the time of finding the target and the distance from origin to the target in the worst case by n robots among which f are Byzantine fault.

In this paper, our main contribution is a new algorithm framework for solving the Byzantine robot searching problem with (n, f) sufficiently large. Under this framework, we design two specific algorithms to improve the previous upper bounds in [13] when \(f/n\in (0.358,0.382)\cup (0.413,0.5)\). Besides, we also improve the upper bound of S(n, f) for some small (n, f). Specifically, we improve the upper bound of S(6, 2) from 4 to 3.682, and the upper bound of S(3, 1) from 9 to 8.53.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Albers, S., Kursawe, K., Schuierer, S.: Exploring unknown environments with obstacles. Algorithmica 32(1), 123–143 (2002)

    Article  MathSciNet  Google Scholar 

  2. Beck, A.: On the linear search problem. Israel J. Math. 2(4), 221–228 (1964)

    Article  MathSciNet  Google Scholar 

  3. Beck, A.: More on the linear search problem. Israel J. Math. 3(2), 61–70 (1965)

    Article  MathSciNet  Google Scholar 

  4. Beck, A., Newman, D.J.: Yet more on the linear search problem. Israel J. Math. 8(4), 419–429 (1970)

    Article  MathSciNet  Google Scholar 

  5. Beck, A., Warren, P.: The return of the linear search problem. Israel J. Math. 14(2), 169–183 (1973)

    Article  MathSciNet  Google Scholar 

  6. Bellman, R.: An optimal search. SIAM Rev. 5(3), 274 (1963)

    Article  Google Scholar 

  7. Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. Parallel Emergent Distrib. Syst. 27(5), 387–408 (2012)

    Article  Google Scholar 

  8. Chrobak, M., Gąsieniec, L., Gorry, T., Martin, R.: Group search on the line. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015. LNCS, vol. 8939, pp. 164–176. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46078-8_14

    Chapter  Google Scholar 

  9. Chuangpishit, H., Czyzowicz, J., Kranakis, E., Krizanc, D.: Rendezvous on a line by location-aware robots despite the presence of byzantine faults. In: Fernández Anta, A., Jurdzinski, T., Mosteiro, M.A., Zhang, Y. (eds.) ALGOSENSORS 2017. LNCS, vol. 10718, pp. 70–83. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72751-6_6

    Chapter  Google Scholar 

  10. Czyzowicz, J., Gasieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Taleb, N.: When patrolmen become corrupted: monitoring a graph using faulty mobile robots. Algorithmica 79(3), 925–940 (2017)

    Article  MathSciNet  Google Scholar 

  11. Czyzowicz, J., et al.: Search on a line by byzantine robots. In: 27th International Symposium on Algorithms and Computation, vol. 27 (2016)

    Google Scholar 

  12. Czyzowicz, J., Georgiou, K., Kranakis, E.: Group search and evacuation. In: Flocchini, P., Prencipe, G., Santoro, N. (eds.) Distributed Computing by Mobile Entities, pp. 335–370. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11072-7_14

    Chapter  Google Scholar 

  13. Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J.: Search on a line with faulty robots. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, pp. 405–414. ACM (2016)

    Google Scholar 

  14. Dieudonné, Y., Pelc, A., Peleg, D.: Gathering despite mischief. ACM Trans. Algorithms (TALG) 11(1), 1 (2014)

    Article  MathSciNet  Google Scholar 

  15. Gal, S.: Minimax solutions for linear search problems. SIAM J. Appl. Math. 27(1), 17–30 (1974)

    Article  MathSciNet  Google Scholar 

  16. Hoffmann, F., Icking, C., Klein, R., Kriegel, K.: The polygon exploration problem. SIAM J. Comput. 31(2), 577–600 (2001)

    Article  MathSciNet  Google Scholar 

  17. Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. Inf. Comput. 131(1), 63–79 (1996)

    Article  MathSciNet  Google Scholar 

  18. Kleinberg, J.M.: On-line search in a simple polygon. In: SODA, vol. 94, pp. 8–15 (1994)

    Google Scholar 

  19. Kupavskii, A., Welzl, E.: Lower bounds for searching robots, some faulty. In: Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pp. 447–453. ACM (2018)

    Google Scholar 

  20. Lamport, L., Shostak, R., Pease, M.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. (TOPLAS) 4(3), 382–401 (1982)

    Article  Google Scholar 

  21. Lynch, N.A.: Distributed Algorithms. Elsevier, Amsterdam (1996)

    MATH  Google Scholar 

  22. Schuierer, S.: Lower bounds in on-line geometric searching. Comput. Geom. 18(1), 37–53 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun, Yuan Sun & Jialin Zhang

  2. University of Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun, Yuan Sun & Jialin Zhang

Authors
  1. Xiaoming Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jialin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoming Sun .

Editor information

Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. University of Massachusetts Lowell, Lowell, MA, USA

    Jie Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Sun, X., Sun, Y., Zhang, J. (2020). Better Upper Bounds for Searching on a Line with Byzantine Robots. In: Du, DZ., Wang, J. (eds) Complexity and Approximation. Lecture Notes in Computer Science(), vol 12000. Springer, Cham. https://doi.org/10.1007/978-3-030-41672-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-41672-0_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41671-3

  • Online ISBN: 978-3-030-41672-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation