Advertisement

Online Graph Exploration with Advice

  • Stefan Dobrev
  • Rastislav Královič
  • Euripides Markou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7355)

Abstract

We study the problem of exploring an unknown undirected graph with non-negative edge weights. Starting at a distinguished initial vertex s, an agent must visit every vertex of the graph and return to s. Upon visiting a node, the agent learns all incident edges, their weights and endpoints. The goal is to find a tour with minimal cost of traversed edges. This variant of the exploration problem has been introduced by Kalyanasundaram and Pruhs in [18] and is known as a fixed graph scenario. There have been recent advances by Megow, Mehlhorn, and Schweitzer ([19]), however the main question whether there exists a deterministic algorithm with constant competitive ratio (w.r.t. to offline algorithm knowing the graph) working on all graphs and with arbitrary edge weights remains open. In this paper we study this problem in the context of advice complexity, investigating the tradeoff between the amount of advice available to the deterministic agent, and the quality of the solution. We show that Ω(n logn) bits of advice are necessary to achieve a competitive ratio of 1 (w.r.t. an optimal algorithm knowing the graph topology). Furthermore, we give a deterministic algorithm which uses O(n) bits of advice and achieves a constant competitive ratio on any graph with arbitrary weights. Finally, going back to the original problem, we prove a lower bound of 5/2 − ε for deterministic algorithms working with no advice, improving the best previous lower bound of 2 − ε of Miyazaki, Morimoto, and Okabe from [20]. In this case, significantly more elaborate technique was needed to achieve the result.

Keywords

Minimum Span Tree Travel Salesman Problem Travel Salesman Problem Competitive Ratio Deterministic 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asahiro, Y., Miyano, E., Miyazaki, S., Yoshimuta, T.: Weighted nearest neighbor algorithms for the graph exploration problem on cycles. Information Processing Letters 110(3), 93–98 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R.: On the Advice Complexity of the k-Server Problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 207–218. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the Advice Complexity of Online Problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 331–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis, vol. 2. Cambridge University Press (1998)Google Scholar
  5. 5.
    Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput. 72(2), 132–143 (2012)CrossRefGoogle Scholar
  6. 6.
    Dobrev, S., Královič, R., Pardubská, D.: Measuring the problem-relevant information in input. ITA 43(3), 585–613 (2009)zbMATHGoogle Scholar
  7. 7.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 412(24), 2642–2656 (2011)zbMATHCrossRefGoogle Scholar
  8. 8.
    Flocchini, P., Mans, B., Santoro, N.: Sense of direction in distributed computing. Theor. Comput. Sci. 291(1), 29–53 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed computing with advice: information sensitivity of graph coloring. Distributed Computing 21(6), 395–403 (2009)CrossRefGoogle Scholar
  10. 10.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Impact of memory size on graph exploration capability. Discrete Applied Mathematics 156(12), 2310–2319 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci. 76(3-4), 222–232 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fraigniaud, P., Korman, A., Lebhar, E.: Local mst computation with short advice. Theory Comput. Syst. 47(4), 920–933 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fusco, E.G., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica 60(4), 719–734 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gutin, G., Punnen, A.P.: The Traveling Salesman Problem and Its Variations. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  15. 15.
    Hromkovič, J., Královič, R., Královič, R.: Information Complexity of Online Problems. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 24–36. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Hurkens, C.A., Woeginger, G.J.: On the nearest neighbor rule for the traveling salesman problem. Operations Research Letters 32(1), 1–4 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ilcinkas, D., Kowalski, D.R., Pelc, A.: Fast radio broadcasting with advice. Theor. Comput. Sci. 411(14-15), 1544–1557 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kalyanasundaram, B., Pruhs, K.R.: Constructing competitive tours from local information. Theoretical Computer Science 130(1), 125–138 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Megow, N., Mehlhorn, K., Schweitzer, P.: Online Graph Exploration: New Results on Old and New Algorithms. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 478–489. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Miyazaki, S., Morimoto, N., Okabe, Y.: The online graph exploration problem on restricted graphs. IEICE Transactions on Information and Systems 92(9), 1620–1627 (2009)CrossRefGoogle Scholar
  21. 21.
    Rosenkrantz, D.J., Stearns, R.E., Lewis II, P.M.: An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput. 6(3), 563–581 (1977)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Dobrev
    • 1
  • Rastislav Královič
    • 2
  • Euripides Markou
    • 3
  1. 1.Institute of MathematicsSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Department of Computer ScienceComenius UniversityBratislavaSlovakia
  3. 3.Department of Computer Science & Biomedical InformaticsUniversity of Central GreeceLamiaGreece

Personalised recommendations