Abstract
The graph search problem is the problem of searching a graph G for a mobile evader by mobile searchers. The edge search is an offline and centralized version, and es(G) denotes the number of searchers necessary and sufficient to edge search G. An online and distributed setting assumes a port numbering of G, a distinct homebase and a whiteboard in each node. Search algorithms typically respect the monotone and connected search strategy to protect the information on whiteboards; however, \(\varOmega ( \frac{n}{\log n} es (G))\) searchers are necessary even for trees, where n is the order of G. We investigate the problem under a new online and distributed setting: We assume that searchers can exchange information wherever they meet, instead of assuming a port numbering, a homebase and whiteboards. Under this setting, we propose a search algorithm for es(G) searchers, which is optimal.
This work was supported in part by Grant-in-Aids for Scientific Research on Innovative Areas “Molecular Robotics” (24104003 and 15H00821) of the Ministry of Education, Culture, Sports, Science, and Technology, Japan, Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” (24106005), and JSPS KAKENHI Grants JP15H02666, JP15K11987 and JP15K15938.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Actually, the settings that these paper adapt are slightly different each other.
- 2.
Formally, the nodes are defined to be anonymous. The searchers however can mark the node as the homebase in its whiteboard introduced below.
- 3.
The bushiness b of a polygon P is the minimum number of triangles that share no edges with P over all triangulations of P.
- 4.
Note that \(p_i\) for all \(i = 1, \ldots , f-1\) are integers representing offsets from the 0-th port. However, \(p_f\) is neither an integer nor an offset. It is an (anonymous) door from which it enters \(u_f\).
- 5.
Algorithm SGL is originally proposed as an algorithm to gather some information, e.g., all identifiers, exploring an unknown graph [4]. In this paper, we modify and use it as a gathering algorithm, modification of which is simple.
- 6.
This assumption is not explicit in [4]. However, in order to define the trajectory R(b, v) from parameter b and v, a constant \(q_0\) must be given.
- 7.
This simulation of \(\pi \) may not be monotone, since searcher \(s_i\) needs to travel inside G violating the monotonicity, when the i-th pebble is picked up from a node and placed at a different node.
References
Aleliunas, R., Karp, R.M., Lipton, R.J., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pp. 218–223 (1979)
Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: Proceedings of the Symposium on Parallel Algorithms and Architectures (SPAA), pp. 200–209 (2002)
Blin, L., Fraigniaud, P., Nisse, N., Vial, S.: Distributed chasing of network intruders. In: Flocchini, P., Gasieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 70–84. Springer, Heidelberg (2006). doi:10.1007/11780823_7
Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. SIAM J. Comput. 44(3), 844–867 (2015)
Efrat, A., Guibas, L.J., Har-Peled, S., Lin, D., Mitchell, J., Murali, T.: Sweeping simple polygon with a chain of guards. In: Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 927–936 (2000)
Flocchini, P., Huang, M.J., Luccio, F.L.: Decontaminating chordal rings and tori using mobile agents. Int. J. Found. Comput. Sci. 18(3), 547–563 (2007)
Flocchini, P., Huang, M.J., Luccio, F.L.: Decontamination of hypercubes by mobile agents. Networks 53(3), 167–178 (2008)
Guibas, L., Latombe, J., LaValle, S., Lin, D., Motwani, R.: A visibility-based pursuit-evasion problem. Int. J. Comput. Geom. Appl. 9(5), 471–494 (1999)
Ilcinkas, D., Nisse, N., Soguet, D.: The cost of monotonicity in distributed graph searching. Distrib. Comput. 22, 117–127 (2009)
Kameda, T., Suzuki, I., Yamashita, M.: An alternative proof for the equivalence of \(\infty \)-searcher and 2-searcher. Theor. Comput. Sci. 634, 108–119 (2016)
Koucký, M.: Universal traversal sequences with backtracking. J. Comput. Syst. Sci. 65(4), 717–726 (2002)
LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40(2), 224–245 (1993)
Lee, J.-H., Park, S.-M., Chwa, K.-Y.: Equivalence of search capability among mobile guards with various visibilities. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 484–495. Springer, Heidelberg (2004). doi:10.1007/978-3-540-30140-0_44
Park, S.-M., Lee, J.-H., Chwa, K.-Y.: Visibility-based pursuit-evasion in a polygonal region by a searcher. In: Orejas, F., Spirakis, P.G., Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 456–468. Springer, Heidelberg (2001). doi:10.1007/3-540-48224-5_38
Parsons, T.D.: Pursuit-evasion in a graph. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 426–441. Springer, Heidelberg (1976)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17 (2008)
Suzuki, I., Yamashita, M.: Searching for a mobile intruder in a polygonal region. SIAM J. Comput. 21(5), 863–888 (1992)
Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973–1022. Elsevier (2000)
Yamashita, M., Umemoto, H., Suzuki, I., Kameda, T.: Searching for mobile intruders in a polygonal region by a group of mobile searchers. Algorithmica 31, 208–236 (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Yakami, T., Yamauchi, Y., Kijima, S., Yamashita, M. (2016). Searching for an Evader in an Unknown Graph by an Optimal Number of Searchers. In: Bonakdarpour, B., Petit, F. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2016. Lecture Notes in Computer Science(), vol 10083. Springer, Cham. https://doi.org/10.1007/978-3-319-49259-9_31
Download citation
DOI: https://doi.org/10.1007/978-3-319-49259-9_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49258-2
Online ISBN: 978-3-319-49259-9
eBook Packages: Computer ScienceComputer Science (R0)