Advertisement

Beachcombing on Strips and Islands

  • Evangelos Bampas
  • Jurek Czyzowicz
  • David Ilcinkas
  • Ralf Klasing
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9536)

Abstract

A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot’s maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers’ Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible.

We first consider the online Beachcombers’ Problem, where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as \(S = \inf _{\ell }{\frac{\ell }{t_A(\ell )}}\), where \(t_A(\ell )\) denotes the time when the search of the segment \([0,\ell ]\) is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over \(\ell \in \mathbb {N}^*\) or \(\ell \ge 1\). We prove that the \(\mathtt {LeapFrog}\) algorithm, which was proposed in [Czyzowicz et al., SIROCCO 2014, LNCS 8576, pp. 23–36 (2014)], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting.

For the offline version of the Beachcombers’ Problem, we consider the single-source Beachcombers’ Problem on the cycle, as well as the multi-source Beachcombers’ Problem on the cycle and on the finite segment. For the single-source Beachcombers’ Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the two-source Beachcombers’ Problem on a finite segment. In consequence, by using results from [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3–21 (2014)], we prove that the single-source Beachcombers’ Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the multi-source variant of the Beachcombers’ Problem on the cycle and on the finite segment, we obtain efficient approximation algorithms.

One important contribution of our work is that, in all variants of the offline Beachcombers’ Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3–21 (2014)], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

Keywords

Completion Time Mobile Robot Optimal Schedule Competitive Ratio Online Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29(4), 1164–1188 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Albers, S.: Online algorithms: a survey. Math. Program. 97(1–2), 3–26 (2003)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Albers, S., Schmelzer, S.: Online algorithms - what is it worth to know the future? In: Vöcking, B., Alt, H., Dietzfelbinger, M., Reischuk, R., Scheideler, C., Vollmer, H., Wagner, D. (eds.) Algorithms Unplugged, pp. 361–366. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer Academic Publishers, Dordrecht (2002). vol. 55Google Scholar
  5. 5.
    Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching in the plane. Inf. Comput. 106, 234–234 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Beauquier, J., Burman, J., Clement, J., Kutten, S.: On utilizing speed in networks of mobile agents. In: ACM SIGACT-SIGOPS 2010, pp. 305–314. ACM (2010)Google Scholar
  7. 7.
    Beck, A.: On the linear search problem. Isr. J. Math. 2(4), 221–228 (1964)zbMATHCrossRefGoogle Scholar
  8. 8.
    Bellman, R.: An optimal search problem. Bull. Am. Math. Soc. 62, 270 (1963)Google Scholar
  9. 9.
    Berman, P.: On-line searching and navigation. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms 1996. LNCS, vol. 1442, pp. 232–241. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Chalopin, J., Flocchini, P., Mans, B., Santoro, N.: Network exploration by silent and oblivious robots. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 208–219. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Czyzowicz, J., Gąsieniec, L., Georgiou, K., Kranakis, E., MacQuarrie, F.: The Beachcombers’ problem: walking and searching with mobile robots. In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 23–36. Springer, Heidelberg (2014)Google Scholar
  12. 12.
    Czyzowicz, J., Gasieniec, L., Georgiou, K., Kranakis, E., MacQuarrie, F.: The multi-source Beachcombers problem. In: Gao, J., Efrat, A., Fekete, S.P., Zhang, Y. (eds.) ALGOSENSORS 2014, LNCS 8847. LNCS, vol. 8847, pp. 3–21. Springer, Heidelberg (2015)Google Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Czyzowicz, J., Ilcinkas, D., Labourel, A., Pelc, A.: Worst-case optimal exploration of terrains with obstacles. Inf. Comput. 225, 16–28 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Das, S., Flocchini, P., Kutten, S., Nayak, A., Santoro, N.: Map construction of unknown graphs by multiple agents. Theor. Comput. Sci. 385(1–3), 34–48 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Demaine, E.D., Fekete, S.P., Gal, S.: Online searching with turn cost. Theor. Comput. Sci. 361(2), 342–355 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. In: Foundations of Computer Science, FOCS 1990, pp. 355–361. IEEE (1990)Google Scholar
  18. 18.
    Deng, X., Kameda, T., Papadimitriou, C.H.: How to learn an unknown environment (extended abstract). In: Foundations of Computer Science, FOCS 1991, pp. 298–303. IEEE (1991)Google Scholar
  19. 19.
    Dereniowski, D., Disser, Y., Kosowski, A., Pająk, D., Uznański, P.: Fast collaborative graph exploration. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 520–532. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  20. 20.
    Fleischer, R., Kamphans, T., Klein, R., Langetepe, E., Trippen, G.: Competitive online approximation of the optimal search ratio. SIAM J. Comput. 38(3), 881–898 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Higashikawa, Y., Katoh, N., Langerman, S., Tanigawa, S.: Online graph exploration algorithms for cycles and trees by multiple searchers. J. Comb. Optim. 28, 480–495 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kawamura, A., Kobayashi, Y.: Fence patrolling by mobile agents with distinct speeds. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 598–608. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  25. 25.
    Wang, G., Irwin, M.J., Fu, H., Berman, P., Zhang, W., Porta, T.L.: Optimizing sensor movement planning for energy efficiency. ACM Trans. Sens. Netw. 7(4), 33 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Evangelos Bampas
    • 1
  • Jurek Czyzowicz
    • 2
  • David Ilcinkas
    • 1
  • Ralf Klasing
    • 1
  1. 1.LaBRI, CNRS and University of BordeauxTalenceFrance
  2. 2.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

Personalised recommendations