Graph Embedding through Random Walk for Shortest Paths Problems
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.
KeywordsGraph embedding shortest paths problem
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