Abstract
Moving an autonomous agent through an unknown environment is one of the crucial problems for robotics and network analysis. Therefore, it received a lot of attention in the last decades and was analyzed in many different settings. The graph exploration problem is a theoretical and abstract model, where an algorithm has to decide how an agent, also called explorer, moves through a network with n vertices and m edges such that every point of interest is visited at least once. For its decisions, the knowledge of the algorithm is limited by the perception capacities of the explorer. We look at the fixed-graph scenario proposed by Kalyanasundaram and Pruhs (ICALP, 1993), where the explorer starts at a vertex of the network and sees all reachable vertices, their unique names and their distance from the current position.
Because the algorithm only learns the structure of the graph during computation, it cannot deterministically compute an optimal tour that visits every vertex at least once without prior knowledge. Therefore, we are interested in the amount of crucial a-priori information needed to solve the problem optimally, which we measure in terms of the well-studied model of advice complexity. Here, a deterministic algorithm can at any time access a binary advice tape written beforehand by an oracle that knows the optimal solution, the graph and the behavior of the algorithm. The number of bits read by the algorithm until the end of computation is called the advice complexity.
We look at the graph exploration problem on unknown directed graphs and focus on cyclic solutions. It is known that \(\mathcal {O}(n\log n)\) bits of advice are necessary and sufficient to compute an optimal solution, for general graphs. In this work, we present algorithms with an advice complexity of \(\mathcal {O}(m)\), thus improving the classical bound for sparse graphs.
This work is supported by the German research council (DFG) Research Training Group 2236 UnRAVeL.
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Notes
- 1.
Note that, with each decision, the algorithm influences the new input for the next decision. Thus, strictly speaking, the graph exploration problem is no classical online problem. But the adversary still knows the behavior of the deterministic algorithm and can, with this knowledge, prepare the input graph, the unique identifiers for the vertices, and thus the enumeration of the edges. Hence, we can analyze the graph exploration problem using the same methodology as used for online problems.
References
Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29(4), 1164–1188 (2000)
Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: exploring and mapping directed graphs. Inf. Comput. 176(1), 1–21 (2002)
Berman, P.: On-line searching and navigation. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms. LNCS, vol. 1442, pp. 232–241. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0029571
Blum, A., Raghavan, P., Schieber, B.: Navigating in unfamiliar geometric terrain. SIAM J. Comput. 26(1), 110–137 (1997)
Böckenhauer, H.J., Fuchs, J., Unger, W.: The graph exploration problem with advice. CoRR abs/1804.06675 (2018)
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). https://doi.org/10.1007/978-3-642-10631-6_35
Brass, P., Vigan, I., Xu, N.: Improved analysis of a multirobot graph exploration strategy. In: 2014 13th International Conference on Control Automation Robotics & Vision (ICARCV), pp. 1906–1910. IEEE (2014)
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). https://doi.org/10.1007/978-3-642-16926-7_20
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)
Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree exploration with little memory. J. Algorithms 51(1), 38–63 (2004)
Dobrev, S., Královič, R., Markou, E.: Online graph exploration with advice. In: Even, G., Halldórsson, M.M. (eds.) SIROCCO 2012. LNCS, vol. 7355, pp. 267–278. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31104-8_23
Dobrev, S., Královič, R., Pardubská, D.: How much information about the future is needed? In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 247–258. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77566-9_21
Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 412(24), 2642–2656 (2011)
Fleischer, R., Trippen, G.: Exploring an unknown graph efficiently. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 11–22. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_4
Foerster, K.T., Wattenhofer, R.: Lower and upper competitive bounds for online directed graph exploration. Theor. Comput. Sci. 655, 15–29 (2016)
Fraigniaud, P., Ilcinkas, D.: Digraphs exploration with little memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24749-4_22
Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput. 206(11), 1276–1287 (2008)
Gorain, B., Pelc, A.: Deterministic graph exploration with advice. In: 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, Warsaw, Poland, 10–14 July 2017, pp. 132:1–132:14 (2017). https://doi.org/10.4230/LIPIcs.ICALP.2017.132
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). https://doi.org/10.1007/978-3-642-15155-2_3
Kalyanasundaram, B., Pruhs, K.R.: Constructing competitive tours from local information. In: Lingas, A., Karlsson, R., Carlsson, S. (eds.) ICALP 1993. LNCS, vol. 700, pp. 102–113. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56939-1_65
Komm, D.: An Introduction to Online Computation - Determinism, Randomization, Advice. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-42749-2
Komm, D., Královič, R., Královič, R., Smula, J.: Treasure hunt with advice. In: Scheideler, C. (ed.) Structural Information and Communication Complexity. LNCS, vol. 9439, pp. 328–341. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25258-2_23
Kortenkamp, D., Bonasso, R.P., Murphy, R.: Artificial Intelligence and Mobile Robots: Case Studies of Successful Robot Systems. MIT Press, Cambridge (1998)
Královič, R.: Personal communication (2017)
Megow, N., Mehlhorn, K., Schweitzer, P.: Online graph exploration: new results on old and new algorithms. Theor. Comput. Sci. 463, 62–72 (2012)
Seibert, S., Sprock, A., Unger, W.: Advice complexity of the online coloring problem. In: Spirakis, P.G., Serna, M. (eds.) CIAC 2013. LNCS, vol. 7878, pp. 345–357. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38233-8_29
Thrun, S., et al.: Autonomous exploration and mapping of abandoned mines. IEEE Robot. Autom. Mag. 11(4), 79–91 (2004)
Thrun, S., et al.: Robotic mapping: a survey. In: Exploring Artificial Intelligence in the New Millennium, vol. 1, pp. 1–35 (2002)
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Böckenhauer, HJ., Fuchs, J., Unger, W. (2018). Exploring Sparse Graphs with Advice (Extended Abstract). In: Epstein, L., Erlebach, T. (eds) Approximation and Online Algorithms. WAOA 2018. Lecture Notes in Computer Science(), vol 11312. Springer, Cham. https://doi.org/10.1007/978-3-030-04693-4_7
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