Abstract
We analyze a randomized pursuit-evasion game on graphs. This game is played by two players, a hunter and a rabbit. Let G be any connected, undirected graph with n nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph G: in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round.
We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for G, the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regards to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only \( \mathcal{O} \) (n log (diam(G))) against restricted as well as unrestricted rabbits. This bound is close to optimal since Ω(n) is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits.
Partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT)
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References
R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovasz, and C. Rackoff. Random walks, universal traversal sequences, and the complexity of maze problems. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science (FOCS), pages 218–223, 1979.
I. Chatzigiannakis, S. Nikoletseas, N. Paspallis, P. Spirakis, and C. Zaroliagis. An experimental study of basic communication protocols in ad-hoc mobile networks. In Proceedings of the 5th Workshop on Algorithmic Engineering, pages 159–171.
I. Chatzigiannakis, S. Nikoletseas, and P. Spirakis. Self-organizing ad-hoc mobile networks: The problem of end-to-end communication. In Proceedings of the 20th ACM Symposium on Principles of Distributed Computing (PODC 2001), 2001.
A. Efrat, L. J. Guibas, S. Har-Peled, D. C. Lin, J. S. B. Mitchell, and T. M. Murali. Sweeping simple polygons with a chain of guards. In Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 927–936, 2000.
M. K. Franklin, Z. Galil, and M. Yung. Eavesdropping games: a graph-theoretic approach to privacy in distributed systems. Journal of the ACM, 47(2):225–243, 2000.
L. J. Guibas, J.-C. Latombe, S. M. LaValle, D. Lin, and R. Motwani. A visibility-based pursuit-evasion problem. International Journal of Computational Geometry and Applications (IJCGA), 9(4):471–493, 1999.
K. Hatzis, G. Pentaris, P. Spirakis, and V. Tampakas. Implementation and testing eavesdropper protocols using the DSP tool. In Proceedings of the 2nd Workshop on Algorithm Engineering (WAE), pages 74–85, 1998.
L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47:205–218, 1986.
A. S. LaPaugh. Recontamination does not help to search a graph. Journal of the ACM, 40(2):224–245, 1993.
S. M. LaValle and J. Hinrichsen. Visibility-based pursuit-evasion: The case of curved environments. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages 1677–1682, 1999.
N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson, and C. H. Papadimitriou. The complexity of searching a graph. Journal of the ACM, 35(1): 18–44, 1988.
R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
S.-M. Park, J.-H. Lee, and K.-Y. Chwa. Visibility-based pursuit-evasion in a polgonal region by a searcher. In Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP), pages 456–468, 2001.
T. D. Parsons. Pursuit-evasion in a graph. In Y Alavi and D. Lick, editors, Theory and Applications of Graphs, Lecture Notes in Mathematics, pages 426–441. Springer, 1976.
T. D. Parsons. The search number of a connected graph. In Proceedings of the 9th Southeastern Conference on Combinatorics, Graph Theory and Computing, pages 549–554, 1978.
I. Suzuki and M. Yamashita. Searching for a mobile intruder in a polygonal region. SIAM Journal on Computing, 21(5):863–888, 1992.
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Adler, M., Räcke, H., Sivadasan, N., Sohler, C., Vöcking, B. (2002). Randomized Pursuit-Evasion in Graphs. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds) Automata, Languages and Programming. ICALP 2002. Lecture Notes in Computer Science, vol 2380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45465-9_77
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DOI: https://doi.org/10.1007/3-540-45465-9_77
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