Advertisement

Patrolling a Path Connecting a Set of Points with Unbalanced Frequencies of Visits

  • Huda Chuangpishit
  • Jurek Czyzowicz
  • Leszek Gąsieniec
  • Konstantinos Georgiou
  • Tomasz Jurdziński
  • Evangelos Kranakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10706)

Abstract

Patrolling consists of scheduling perpetual movements of a collection of mobile robots, so that each point of the environment is regularly revisited by any robot in the collection. In previous research, it was assumed that all points of the environment needed to be revisited with the same minimal frequency.

In this paper we study efficient patrolling protocols for points located on a path, where each point may have a different constraint on frequency of visits. The problem of visiting such divergent points was recently posed by Gąsieniec et al. in [14], where the authors study protocols using a single robot patrolling a set of n points located in nodes of a complete graph and in Euclidean spaces.

The focus in this paper is on patrolling with two robots. We adopt a scenario in which all points to be patrolled are located on a line. We provide several approximation algorithms concluding with the best currently known \(\sqrt{3}\)-approximation.

References

  1. 1.
    Alshamrani, S., Kowalski, D.R., Gąsieniec, L.: How reduce max algorithm behaves with symptoms appearance on virtual machines in clouds. In: Proceedings of IEEE International Conference CIT/IUCC/DASC/PICOM, pp. 1703–1710 (2015)Google Scholar
  2. 2.
    Baruah, S.K., Cohen, N.K., Plaxton, C.G., Varvel, D.A.: Proportionate progress: a notion of fairness in resource allocation. Algorithmica 15(6), 600–625 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baruah, S.K., Lin, S.-S.: Pfair scheduling of generalized pinwheel task systems. IEEE Trans. Comput. 47(7), 812–816 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bender, M.A., Fekete, S.P., Kröller, A., Mitchell, J.S.B., Liberatore, V., Polishchuk, V., Suomela, J.: The minimum backlog problem. Theoret. Comput. Sci. 605, 51–61 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, M.H.L., Hurkens, C.A.J., Kusters, V.J.J., Staals, F., Woeginger, G.J., Zantema, H.: Cinderella versus the wicked stepmother. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 57–71. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-33475-7_5 CrossRefGoogle Scholar
  6. 6.
    Chan, M.Y., Chin, F.Y.L.: General schedulers for the pinwheel problem based on double-integer reduction. IEEE Trans. Comput. 41(6), 755–768 (1992)CrossRefGoogle Scholar
  7. 7.
    Chan, M.Y., Chin, F.: Schedulers for larger classes of pinwheel instances. Algorithmica 9(5), 425–462 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chuangpishit, H., Czyzowicz, J., Gasieniec, L., Georgiou, K., Jurdzinski, T., Kranakis, E.: Patrolling a path connecting set of points with unbalanced frequencies of visits (2012). http://arxiv.org/abs/1710.00466
  9. 9.
    Chrobak, M., Csirik, J., Imreh, C., Noga, J., Sgall, J., Woeginger, G.J.: The buffer minimization problem for multiprocessor scheduling with conflicts. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 862–874. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-48224-5_70 CrossRefGoogle Scholar
  10. 10.
    Collins, A., Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Martin, R., Morales Ponce, O.: Optimal patrolling of fragmented boundaries. In: Proceedings of the Twenty-fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2013, New York, USA, pp. 241–250 (2013)Google Scholar
  11. 11.
    Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E.: Boundary patrolling by mobile agents with distinct maximal speeds. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 701–712. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23719-5_59 CrossRefGoogle Scholar
  12. 12.
    Czyzowicz, J., Gasieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Taleb, N.: When patrolmen become corrupted: monitoring a graph using faulty mobile robots. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 343–354. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48971-0_30 CrossRefGoogle Scholar
  13. 13.
    Fishburn, P.C., Lagarias, J.C.: Pinwheel scheduling: achievable densities. Algorithmica 34(1), 14–38 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gąsieniec, L., Klasing, R., Levcopoulos, C., Lingas, A., Min, J., Radzik, T.: Bamboo garden trimming problem (perpetual maintenance of machines with different attendance urgency factors). In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds.) SOFSEM 2017. LNCS, vol. 10139, pp. 229–240. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-51963-0_18 CrossRefGoogle Scholar
  15. 15.
    Holte, R., Mok, A., Rosier, L., Tulchinsky, I., Varvel, D.: The pinwheel: a real-time scheduling problem. In: II: Software Track, Proceedings of the Twenty-Second Annual Hawaii International Conference on System Sciences, vol. 2, pp. 693–702, January 1989Google Scholar
  16. 16.
    Holte, R., Rosier, L., Tulchinsky, I., Varvel, D.: Pinwheel scheduling with two distinct numbers. Theoret. Comput. Sci. 100(1), 105–135 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kawamura, A., Kobayashi, Y.: Fence patrolling by mobile agents with distinct speeds. Distrib. Comput. 28(2), 147–154 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liang, D., Shen, H.: Point sweep coverage on path. Unpublished work https://arxiv.org/abs/1704.04332
  19. 19.
    Lin, S.-S., Lin, K.-J.: A pinwheel scheduler for three distinct numbers with a tight schedulability bound. Algorithmica 19(4), 411–426 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ntafos, S.: On gallery watchmen in grids. Inf. Process. Lett. 23(2), 99–102 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    O’Rourke, J.: Art Gallery Theorems and Algorithms, vol. 57. Oxford University Press, Oxford (1987)zbMATHGoogle Scholar
  22. 22.
    Romer, T.H., Rosier, L.E.: An algorithm reminiscent of Euclidean-gcd for computing a function related to pinwheel scheduling. Algorithmica 17(1), 1–10 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Serafini, P., Ukovich, W.: A mathematical model for periodic scheduling problems. SIAM J. Discret. Math. 2(4), 550–581 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Urrutia, J.: Art gallery and illumination problems. Handbook Comput. Geom. 1(1), 973–1027 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Huda Chuangpishit
    • 1
  • Jurek Czyzowicz
    • 2
  • Leszek Gąsieniec
    • 3
  • Konstantinos Georgiou
    • 1
  • Tomasz Jurdziński
    • 4
  • Evangelos Kranakis
    • 5
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada
  2. 2.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada
  3. 3.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  4. 4.Instytut InformatykiUniwersytet WrocławskiWrocławPoland
  5. 5.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations