Multiple source shortest paths in a genus g graph

We give an <i>O</i>(<i>g</i><sup>2</sup><i>n</i>log <i>n</i>) algorithm to represent the shortest path tree from all the vertices on a single specified face <i>f</i> in a genus <i>g</i> graph. From this representation, any query distance from a vertex in <i>f</i> can be obtained in <i>O</i>(log <i>n</i>) time. The algorithm uses a kinetic data structure, where the source of the tree iteratively moves across edges in <i>f</i>. In addition, we give applications using these shortest path trees in order to compute the shortest non-contractible cycle and the shortest non-separating cycle embedded on an orientable 2-manifold in <i>O</i>(<i>g</i><sup>3</sup><i>n</i>log <i>n</i>) time.

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