Showing papers by "Jeff Erickson published in 2008"

Splitting (complicated) surfaces is hard

Abstract: Let M be an orientable combinatorial surface. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither of which is homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus g and the number of boundary components b of the surface. Specifically, we describe an algorithm to compute the shortest splitting cycle in (g+b)^O^(^g^+^b^)nlogn time, where n is the complexity of the combinatorial surface.

Walking your dog in the woods in polynomial time

Abstract: The Frechet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Frechet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles ("trees"). We describe a polynomial-time algorithm to compute the homotopic Frechet distance between two given polygonal curves in the plane minus a given set of obstacles, which are either points or polygons.

Testing contractibility in planar rips complexes

Abstract: The (Vietoris-)Rips complex of a discrete point-set P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest non-contractible cycle in a planar Rips complex in O(n2log n + mn) time.

Empty-ellipse graphs

Abstract: We define and study a geometric graph over points in the plane that captures the local behavior of Delaunay triangulations of points on smooth surfaces in IR3. Two points in a planar point set P are neighbors in the empty-ellipse graph if they lie on an axis-aligned ellipse with no point of P in its interior. The empty-ellipse graph can be a clique in the worst case, but it is usually much less dense. Specifically, the empty-ellipse graph of n points has complexity Θ(Δn) in the worst case, where Δ is the ratio between the largest and smallest pairwise distances. For points generated uniformly at random in a rectangle, the empty-ellipse graph has expected complexity Θ(n log n). As an application of our proof techniques, we show that the Delaunay triangulation of n random points on a circular cylinder has expected complexity Θ(n log n).

Finding one tight cycle

Abstract: A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class We describe an algorithm to compute a single tight, non-contractible, simple cycle on a given orientable combinatorial surface in O(n log n) time The only method previously known for this problem was to compute the globally shortest non-contractible or non-separating cycle in O(min{g3, n} n log n) time, where g is the genus of the surface As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O(n log n) time and a tight octagonal decomposition in O(gn log n) time

Computing interesting topological features

Abstract: Many questions about homotopy are provably hard or even unsolvable in general. However, in specific settings, it is possible to efficiently test homotopy-equivalence or compute shortest cycles with prescribed homotopy. We focus on computing such "interesting" topological features in three settings. The first two results are about cycles on surfaces; the third is about classes of homotopies in R2 minus a set of obstacles; and the final result is about paths and cycles in Rips complexes. First, we examine two problems in the combinatorial surface model. Combinatorial surfaces combine properties of graphs and manifolds, making a rich set of techniques available for analysis and algorithm design. We give algorithms to find the shortest noncontractible and nonseparating cycles in a combinatorial surface in O(g3n log n) time. Our main tool is a data structure that kinetically maintains the shortest path tree as the root of the tree moves around the vertices of a single face. The total running time is O(g2n log n). By maintaining the data structure persistently, we can answer shortest path queries in O(log n) time. Next we consider finding the shortest splitting cycle in a combinatorial surface, or simple cycle which is both separating and noncontractible; such cycles divide the topology of the surface as well as the underlying graph. We prove that finding the shortest splitting cycle is NP-Hard. We then give an algorithm that runs gO(g) n log n time, which is polynomial if the surface is fixed. We then examine a very different setting, namely similarity between curves in some underlying metric space. If we imagine a homotopy between the curves as a way to morph one curve into the other, we can optimize the morphing so that the maximum distance any point must travel is minimized This is a generalization of the more well known Frechet distance, with the additional requirement that the leash to move continuously in the ambient space. We call this distance the homotopic Frechet distance. We give a polynomial time algorithm to compute the homotopic Frechet distance between two curves in the plane minus a set of polygonal obstacles. We also extend our characterization of optimal morphings to surfaces of nonpositive curvature. Finally, we examine a more fundamental homotopy problem in a different setting. A Rips complex is a simplicial complex defined by a set of points from some metric space where every pair of points within distance 1 is connected by an edge, and every (k + 1)-clique in that graph forms a k-simplex. We prove that the projection map which takes each k-simplex in the Rips complex to the convex hull of the original points in the plane induces an isomorphism between the fundamental groups of both spaces. Since the union of these convex hulls is a polygonal region in the plane, possibly with holes, our result implies that the fundamental group of a planar Rips complex is a free group, allowing us to design efficient algorithms to answer homotopy questions in planar Rips complexes.