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.

We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G ∗. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously increase the parameter, each change in the shortest path tree can be effected in O(log n) time using standard dynamic tree data structures, and the special structure of the parametrization implies that each directed edge enters the evolving shortest path tree at most once. The resulting maximum-flow algorithm is identical to the recent algorithm of Borradaile and Klein [J. ACM 2009], but our new formulation allows a simpler presentation and analysis. On the other hand, we demonstrate that for a similarly structured parametric shortest path problem on the torus, the shortest path tree can change Ω(n²) times in the worst case, suggesting that a different method may be required to efficiently compute maximum flows in higher-genus graphs.

We consider the problem of finding all Pareto-optimal (s, t)-paths in discrete time-dependent graphs with respect to several criteria. Label setting or label correcting algorithms can be used to compute all Pareto-optima in static multi-dimensional weighted digraphs. These are generalizations of the wellknown algorithms for the single criterion shortest paths problem. Their correctness is based on the subpath-optimality. Unfortunately, this property vanishes in the classical notions of time-dependent networks. Therefore, Kostreva et al. (1993), Getachew et al. (2000) and Hamacher et al. (2006) solved the multi-criteria time-dependent problem only for special cases. We give some new definitions for discrete time-dependent graphs which differ in some aspects from previous works. These new notions allow us to establish the subpath-optimality in discrete time-dependent graphs. Our concept can be interpreted as an implicit time expansion. Furthermore, we extend this concept to event-dependent graphs. This is useful for many practical applications in which weights of paths do not only depend on time but also on other attributes.