Practical methods for approximating shortest paths on a convex polytope in R3

AbstractWe propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p, q on its surface, let dP(p, q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2dP(p, q) in time O(n). Extending this result, we can also compute an approximation of the shortest path tree rooted at an arbitrary point x ∈ P in time O(n log n). In the approximate tree, the distance between a vertex v ∈ P and x is at most cdP(x, v), where c = 2.38(1 + ε) for any fixed ε > 0. The best algorithms for computing an exact shortest path on a convex polytope take Ω(n2) time in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in O(n) time our algorithm gives a path of length at most 2k times the optimal

How to Walk Your Dog in the Mountains with No Magic Leash

We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic \Frechet distance, or the minimum height of a homotopy, is in NP

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

An Exponential Lower Bound on the Complexity of Regularization Paths

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal parameter much easier. It has been assumed that these piecewise linear solution paths have only linear complexity, i.e. linearly many bends. We prove that for the support vector machine this complexity can be exponential in the number of training points in the worst case. More strongly, we construct a single instance of n input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) = \Theta(2^d) many distinct subsets of support vectors occur as the regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure

Geodesic Paths On 3D Surfaces: Survey and Open Problems

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces

Exact Geosedics and Shortest Paths on Polyhedral Surface

We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.National Institute for Biomedical Imaging and Bioengineering (R01 EB001550