A critical point for random graphs with a given degree sequence
Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
Computing almost shortest paths
Proceedings of the twentieth annual ACM symposium on Principles of distributed computing
Efficient Routing in Networks with Long Range Contacts
DISC '01 Proceedings of the 15th International Conference on Distributed Computing
Analyzing Kleinberg's (and other) small-world Models
Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing
Universal augmentation schemes for network navigability: overcoming the √n-barrier
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Polylogarithmic network navigability using compact metrics with small stretch
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Witnesses for Boolean matrix multiplication and for shortest paths
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Two new graphs kernels in chemoinformatics
Pattern Recognition Letters
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We present a new probabilistic technique of embedding graphs in Zd, 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 Zd. 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 nonsparse 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.