Abstract
We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent.
We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k = D n 1 + ε < n 2 + ε, for any ε > 0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D.
We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large k. For any constant c > 1, we show that in the global communication model, a team of k = D n c agents can always complete exploration in \(D(1+ \frac{1}{c-1} +o(1))\) time steps, whereas at least \(D(1+ \frac{1}{c} -o(1))\) steps are sometimes required. In the local communication model, \(D(1+ \frac{2}{c-1} +o(1))\) steps always suffice to complete exploration, and at least \(D(1+ \frac{2}{c} -o(1))\) steps are sometimes required. This shows a clear separation between the global and local communication models.
This work was initiated while A. Kosowski was visiting Y. Disser at ETH Zurich. Supported by ANR project DISPLEXITY and by NCN under contract DEC-2011/02/A/ST6/00201. The authors are grateful to Shantanu Das for valuable discussions and comments on the manuscript. The full version of this paper is available online at: http://hal.inria.fr/hal-00802308 .
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Awerbuch, B., Betke, M., Rivest, R.L., Singh, M.: Piecemeal graph exploration by a mobile robot. Information and Computation 152(2), 155–172 (1999)
Brass, P., Cabrera-Mora, F., Gasparri, A., Xiao, J.: Multirobot tree and graph exploration. IEEE Transactions on Robotics 27(4), 707–717 (2011)
Czyzowicz, J., Ilcinkas, D., Labourel, A., Pelc, A.: Worst-case optimal exploration of terrains with obstacles. Information and Computation 225, 16–28 (2013)
Duncan, C.A., Kobourov, S.G., Kumar, V.S.A.: Optimal constrained graph exploration. ACM Transactions on Algorithms 2(3), 380–402 (2006)
Dynia, M., Korzeniowski, M., Schindelhauer, C.: Power-aware collective tree exploration. In: Grass, W., Sick, B., Waldschmidt, K. (eds.) ARCS 2006. LNCS, vol. 3894, pp. 341–351. Springer, Heidelberg (2006)
Dynia, M., Kutyłowski, J., Meyer auf der Heide, F., Schindelhauer, C.: Smart robot teams exploring sparse trees. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 327–338. Springer, Heidelberg (2006)
Dynia, M., Łopuszański, J., Schindelhauer, C.: Why robots need maps. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 41–50. Springer, Heidelberg (2007)
Fraigniaud, P., Gąsieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)
Frederickson, G.N., Hecht, M.S., Kim, C.E.: Approximation algorithms for some routing problems. SIAM Journal on Computing 7(2), 178–193 (1978)
Gabriely, Y., Rimon, E.: Competitive on-line coverage of grid environments by a mobile robot. Computational Geometry 24(3), 197–224 (2003)
Herrmann, D., Kamphans, T., Langetepe, E.: Exploring simple triangular and hexagonal grid polygons online. CoRR, abs/1012.5253 (2010)
Higashikawa, Y., Katoh, N.: Online exploration of all vertices in a simple polygon. In: Proc. 6th Frontiers in Algorithmics Workshop and the 8th Int. Conf. on Algorithmic Aspects of Information and Management (FAW-AAIM), pp. 315–326 (2012)
Higashikawa, Y., Katoh, N., Langerman, S., Tanigawa, S.-I.: Online graph exploration algorithms for cycles and trees by multiple searchers. Journal of Combinatorial Optimization (2013)
Icking, C., Kamphans, T., Klein, R., Langetepe, E.: Exploring an unknown cellular environment. In: Proc. 16th European Workshop on Computational Geometry (EuroCG), pp. 140–143 (2000)
Kolenderska, A., Kosowski, A., Małafiejski, M., Żyliński, P.: An improved strategy for exploring a grid polygon. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 222–236. Springer, Heidelberg (2010)
Łopuszański, J.: Tree exploration. Tech-report, Institute of Computer Science, University of Wrocław, Poland (2007) (in Polish)
Ortolf, C., Schindelhauer, C.: Online multi-robot exploration of grid graphs with rectangular obstacles. In: Proc. 24th ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pp. 27–36 (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dereniowski, D., Disser, Y., Kosowski, A., Pająk, D., Uznański, P. (2013). Fast Collaborative Graph Exploration. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_46
Download citation
DOI: https://doi.org/10.1007/978-3-642-39212-2_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39211-5
Online ISBN: 978-3-642-39212-2
eBook Packages: Computer ScienceComputer Science (R0)