Multiple-Source Shortest Paths in Embedded Graphs
Abstract
Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(gn log n) time with high probability, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [SODA 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g^2 n log n) time with high probability. Our high-probability time bounds hold in the worst-case for generic edge weights, or with an additional O(log n) factor for arbitrary edge weights.
Cite
@article{arxiv.1202.0314,
title = {Multiple-Source Shortest Paths in Embedded Graphs},
author = {Sergio Cabello and Erin Wolf Chambers and Jeff Erickson},
journal= {arXiv preprint arXiv:1202.0314},
year = {2013}
}
Comments
31 pages, 3 figures. Accepted to SIAM Journal on Computing. Preliminary version, without the third author's contributions, in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 89-97, 2007. In this revision we provide a careful treatment of non-generic weights