Showing papers by "Jeff Erickson published in 2010"

Homotopic Fréchet distance between curves or, 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 polygonal obstacles.

Vietoris–Rips Complexes of Planar Point Sets

Abstract: Fix a finite set of points in Euclidean n-space \(\mathbb{E}^{n}\) , thought of as a point-cloud sampling of a certain domain \(D\subset\mathbb{E}^{n}\) . The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to \(\mathbb{E}^{n}\) that has as its image a more accurate n-dimensional approximation to the homotopy type of D.

Maximum flows and parametric shortest paths in planar graphs

Abstract: 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.

Tightening Nonsimple Paths and Cycles on Surfaces

Abstract: We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity $n$, genus $g\geq2$, and no boundary, we construct in $O(gn\log n)$ time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity $k$ in $O(gnk)$ time, or the shortest cycle homotopic to a given cycle of complexity $k$ in $O(gnk\log(nk))$ time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdiere and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.

Computing the Shortest Essential Cycle

Abstract: An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2log n) time, or in O(nlog n) time when both the genus and number of boundaries are fixed. Our results correct an error in a paper of Erickson and Har-Peled (Discrete Comput. Geom. 31(1):37–59, 2004).

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, noncontractible, essentially 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 noncontractible or nonseparating 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, a tight octagonal decomposition in O(gnlog n) time, and a shortest contractible cycle enclosing a nonempty set of faces in O(nlog2n) time.

Well-centered meshing

Abstract: A well-centered simplex is a simplex whose circumcenter lies in its interior, and a well-centered mesh is a simplicial mesh in which every simplex is well-centered. We examine properties of the well-centered simplex and well-centered meshes, present experimental results from an optimization method designed to make meshes well-centered, and give examples of well-centered tetrahedral meshes of a variety of three-dimensional regions.