## Multiplesource shortest paths in embedded graphs (2012)

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Citations: | 7 - 5 self |

### BibTeX

@MISC{Cabello12multiplesourceshortest,

author = {Sergio Cabello and Erin W. Chambers and Jeff Erickson},

title = {Multiplesource shortest paths in embedded graphs},

year = {2012}

}

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### 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 an algorithm to preprocess the graph in O(gn log n) time, 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 [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 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(g² n log n) time.