Abstract
In this paper, we deal with an error model in distributed networks. For a target t, every node is assumed to give an advice, ie. to point to a neighbour that take closer to the destination. Any node giving a bad advice is called a liar. Starting from a situation without any liar, we study the impact of topology changes on the number of liars.
More precisely, we establish a relationship between the number of liars and the number of distance changes after one edge deletion. Whenever ℓ deleted edges are chosen uniformly at random, for any graph with n nodes, m edges and diameter D, we prove that the expected number of liars and distance changes is \(O(\frac{\ell^2Dn}{m})\) in the resulting graph. The result is tight for ℓ = 1. For some specific topologies, we give more precise bounds.
This work is granted by the european project EULER.
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References
Bernstein, A.: Fully dynamic (2 + epsilon) approximate all-pairs shortest paths with fast query and close to linear update time. In: FOCS, pp. 693–702. IEEE Computer Society (2009)
Bernstein, A., Karger, D.R.: A nearly optimal oracle for avoiding failed vertices and edges. In: Mitzenmacher, M. (ed.) STOC, pp. 101–110. ACM (2009)
Chung, F.R.K., Garey, M.R.: Diameter bounds for altered graphs. Journal of Graph Theory 8(4), 511–534 (1984)
Chechik, S., Langberg, M., Peleg, D., Roditty, L.: f-Sensitivity Distance Oracles and Routing Schemes. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 84–96. Springer, Heidelberg (2010)
Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)
Demetrescu, C., Thorup, M., Chowdhury, R.A., Ramachandran, V.: Oracles for distances avoiding a failed node or link. SIAM J. Comput. 37, 1299–1318 (2008)
Hanusse, N., Ilcinkas, D., Kosowski, A., Nisse, N.: Locating a Target with an Agent Guided by Unreliable Local Advice. In: Proceedings of the 29th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing PODC 2010, Zurich, Suisse, pp. 355–364. ACM, New York (2010)
Hanusse, N., Kranakis, E., Krizanc, D.: Searching with mobile agents in networks with liars. Discrete Applied Mathematics 137, 69–85 (2004)
Hanusse, N., Kavvadias, D.J., Kranakis, E., Krizanc, D.: Memoryless search algorithms in a network with faulty advice. Theor. Comput. Sci. 402(2-3), 190–198 (2008)
Hershberger, J., Suri, S.: Vickrey prices and shortest paths: What is an edge worth? In: FOCS, pp. 252–259 (2001)
Khanna, N., Baswana, S.: Approximate shortest paths avoiding a failed vertex: Optimal size data structures for unweighted graphs. In: Marion, J.-Y., Schwentick, T. (eds.) STACS. LIPIcs, pp. 513–524. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)
King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: FOCS, pp. 81–91 (1999)
Kranakis, E., Krizanc, D.: Searching with uncertainty. In: Proc. SIROCCO 1999, pp. 194–203 (1999)
Nardelli, E., Proietti, G., Widmayer, P.: Finding the most vital node of a shortest path. Theor. Comput. Sci. 296, 167–177 (2003)
Schoone, A., Bodlaender, H., van Leeuwen, J.: Improved Diameter Bounds for Altered Graphs. In: Tinhofer, G., Schmidt, G. (eds.) WG 1986. LNCS, vol. 246, pp. 227–236. Springer, Heidelberg (1987)
Thorup, M.: Fully-Dynamic All-Pairs Shortest Paths: Faster and Allowing Negative Cycles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 384–396. Springer, Heidelberg (2004)
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Glacet, C., Hanusse, N., Ilcinkas, D. (2011). The Impact of Edge Deletions on the Number of Errors in Networks. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds) Principles of Distributed Systems. OPODIS 2011. Lecture Notes in Computer Science, vol 7109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25873-2_26
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DOI: https://doi.org/10.1007/978-3-642-25873-2_26
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