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.

Let G be a graph cellularly embedded on a surface. We consider the problem of determining whether G contains a cycle (i.e. a closed walk without repeated vertices) of a certain topological type. We show that the problem can be answered in linear time when the topological type is one of the following: contractible, non-contractible, or non-separating. In either case we obtain the same time complexity if we require the cycle to contain a given vertex. On the other hand, we prove that the problem is NP-complete when considering separating or splitting cycles. We also show that deciding the existence of a separating or a splitting cycle of length at most k is fixed-parameter tractable with respect tok plus the genus of the surface.

We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log 3 n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n 2 log n) time. We can also achieve a deterministic g O(g) n 2 log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus. 1

Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in the application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of high-dimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in two types of features in high dimensions: local branching structures and global circular structures. We construct local and global circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Our results reveal never-before-seen structures on real-world data sets from a variety of applications. Branching and Circular Features in High Dimensional Data

Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest non-contractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest non-null-homologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest non-separating cycle.

We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimum-weight cycles in a given homotopy or homology class, and computing minimum-weight cycle bases for the fundamental group or various homology groups.