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.
E. Markou was supported in part by a research grant offered by the Ministries of Education of Greece and Slovakia (Bilateral Exchange Programme for Researchers) and by THALES: ALGONOW project funded by the Greek Ministry of Education.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
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)
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis, vol. 2. Cambridge University Press (1998)
Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput. 72(2), 132–143 (2012)
Dobrev, S., Královič, R., Pardubská, D.: Measuring the problem-relevant information in input. ITA 43(3), 585–613 (2009)
Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 412(24), 2642–2656 (2011)
Flocchini, P., Mans, B., Santoro, N.: Sense of direction in distributed computing. Theor. Comput. Sci. 291(1), 29–53 (2003)
Fraigniaud, P., Gavoille, C., Ilcinkas, D., Pelc, A.: Distributed computing with advice: information sensitivity of graph coloring. Distributed Computing 21(6), 395–403 (2009)
Fraigniaud, P., Ilcinkas, D., Pelc, A.: Impact of memory size on graph exploration capability. Discrete Applied Mathematics 156(12), 2310–2319 (2008)
Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci. 76(3-4), 222–232 (2010)
Fraigniaud, P., Korman, A., Lebhar, E.: Local mst computation with short advice. Theory Comput. Syst. 47(4), 920–933 (2010)
Fusco, E.G., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica 60(4), 719–734 (2011)
Gutin, G., Punnen, A.P.: The Traveling Salesman Problem and Its Variations. Springer, Heidelberg (2002)
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)
Hurkens, C.A., Woeginger, G.J.: On the nearest neighbor rule for the traveling salesman problem. Operations Research Letters 32(1), 1–4 (2004)
Ilcinkas, D., Kowalski, D.R., Pelc, A.: Fast radio broadcasting with advice. Theor. Comput. Sci. 411(14-15), 1544–1557 (2010)
Kalyanasundaram, B., Pruhs, K.R.: Constructing competitive tours from local information. Theoretical Computer Science 130(1), 125–138 (1994)
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)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dobrev, S., Královič, R., Markou, E. (2012). Online Graph Exploration with Advice. In: Even, G., Halldórsson, M.M. (eds) Structural Information and Communication Complexity. SIROCCO 2012. Lecture Notes in Computer Science, vol 7355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31104-8_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-31104-8_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31103-1
Online ISBN: 978-3-642-31104-8
eBook Packages: Computer ScienceComputer Science (R0)