Multiple-Source Shortest Paths in Embedded Graphs

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(gnlog 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(nlog 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 nlog n) time.

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Sergio Cabello, Erin W. Chambers, Jeff Erickson
• Keywords: CORR 2012, Discrete Algorithms, Education, Planar Graphs, Preprocessing Algorithm