Abstract
We present a new probabilistic technique of embedding graphs in Z d, the d-dimensional integer lattice, in order to find the shortest paths and shortest distances between pairs of nodes. In our method the nodes of a breath first search (BFS) tree, starting at a particular node, are labeled as the sites found by a branching random walk on Z d. After describing a greedy algorithm for routing (distance estimation) which uses the ℓ1 distance (ℓ2 distance) between the labels of nodes, we approach the following question:
Assume that the shortest distance between nodes s and t in the graph is the same as the shortest distance between them in the BFS tree corresponding to the embedding, what is the probability that our algorithm finds the shortest path (distance) between them correctly?
Our key result comprises the following two complementary facts: i) by choosing d ≐ d(n) (where n is the number of nodes) large enough our algorithm is successful with high probability, and ii) d does not have to be very large - in particular it suffices to have d = O( polylog(n) ).
We also suggest an adaptation of our technique to finding an efficient solution for the all-sources all-targets (ASAT) shortest paths problem, using the fact that a single embedding finds not only the shortest paths (distances) from its origin to all other nodes, but also between several other pairs of nodes. We demonstrate its behavior on a specific non-sparse random graph model and on real data, the PGP network, and obtain promising results.
The method presented here is less likely to prove useful as an attempt to find more efficient solutions for ASAT problems, but rather as the basis for a new approach for algorithms and protocols for routing and communication. In this approach, noise and the resulting corruption of data delivered in various channels might actually be useful when trying to infer the optimal way to communicate with distant peers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Galil, Z., Margalit, O., Naor, M.: Witnesses for Boolean matrix multiplication and for shortest paths. In: Proceeedings of the 33rd IEEE Symposium on Foundations of Computer Science, Pittsburgh, PA, pp. 417–426 (1992)
Barriere, L., Fraigniaud, P., Kranakis, E., Krizanc, D.: Efficient Routing in Networks with Long Range Contacts. In: Welch, J.L. (ed.) DISC 2001. LNCS, vol. 2180, pp. 270–284. Springer, Heidelberg (2001)
Boguna, M., Pastor–Satorras, R., Diaz–Guilera, A., Arenas, A.: Models of social networks based on social distance attachment. Phys. Rev. E 70, 056122 (2004)
Elkin, M.: Computing almost shortest paths. In: Proceedings of the 20th ACM Symposium on Principles of Distributed Computing, Newport, RI, August, pp. 53–63 (2001)
Fraigniaud, P., Gavoille, C.: Polylogarithmic Network Navigability Using Compact Metrics with Small Stretch. In: SPAA 2008: Proceedings of the Twentieth Annual Symposium on Parallelism in Algorithms and Architectures, pp. 62–69. ACM, New York (2008)
Fraigniaud, P., Gavoille, C., Kosowski, A., Lebhar, E., Lotker, Z.: Universal Augmentation Schemes for Network Navigability: Overcoming the sqrt(n)–Barrier. In: Proceedings of the Nineteenth Annual ACM Symp. on Parallel Algorithms and Architectures, pp. 1–7. ACM, New York (2007)
Kleinberg, J.M.: Navigation in a small world. Nature 406, 845 (2000)
Lass, H., Gottlieb, P.: Probability and statistics. Addison-Wesley, Reading (1971)
Liben–Nowell, D., Novak, J., Kumar, R., Raghavan, P., Tomkins, A.: Geographical routing in social networks. Proceedings of the National Academy of Science 102, 11623–11628 (2005)
Martel, C., Nguyen, V.: Analyzing kleinberg’s (and other) small-world models. In: PODC 2004: Proceedings of the twentythird annual ACM symposium on Principles of distributed computing, pp. 179–188. ACM Press, New York (2004)
Milgram, S.: The small world problem. Psychology Today 2, 60–67 (1967)
Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence, Random Struct. Algorithms 6, 161–179 (1995)
Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications, Phys. Phys. Rev. E 64, 026118 (2001)
Prohorov, Y.V., Rozanov, Y.A.: Probability theory, basic concepts. Limit theorems, random processes. Springer, Heidelberg (1969) (Translated from Russian)
van den Esker, H., van der Hofstad, R., Hooghiemstra, G., Znamenski, D.: Distances in random graphs with infinite mean degrees. Extremes 8, 111–140 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berchenko, Y., Teicher, M. (2009). Graph Embedding through Random Walk for Shortest Paths Problems. In: Watanabe, O., Zeugmann, T. (eds) Stochastic Algorithms: Foundations and Applications. SAGA 2009. Lecture Notes in Computer Science, vol 5792. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04944-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-04944-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04943-9
Online ISBN: 978-3-642-04944-6
eBook Packages: Computer ScienceComputer Science (R0)