Approximations of the Wiener sausage and its curvature measures

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.Comment: Published in at http://dx.doi.org/10.1214/09-AAP596 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

We provide a characterization of the radii minimal projections of polytopes onto $j$-dimensional subspaces in Euclidean space \E^n. Applied on simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterization to determine the sequence of outer $(n-1)$-radii of regular simplices (which are the radii of smallest enclosing cylinders). This settles a question which arose from the incidence that a paper by Wei{\ss}bach (1983) on this determination was erroneous. In the proof, we first reduce the problem to a constrained optimization problem of symmetric polynomials and then to an optimization problem in a fixed number of variables with additional integer constraints.Comment: Minor revisions. To appear in Advances in Geometr

Hybrid LSH: Faster Near Neighbors Reporting in High-dimensional Space

We study the $r$-near neighbors reporting problem ($r$-NN), i.e., reporting \emph{all} points in a high-dimensional point set $S$ that lie within a radius $r$ of a given query point $q$. Our approach builds upon on the locality-sensitive hashing (LSH) framework due to its appealing asymptotic sublinear query time for near neighbor search problems in high-dimensional space. A bottleneck of the traditional LSH scheme for solving $r$-NN is that its performance is sensitive to data and query-dependent parameters. On datasets whose data distributions have diverse local density patterns, LSH with inappropriate tuning parameters can sometimes be outperformed by a simple linear search. In this paper, we introduce a hybrid search strategy between LSH-based search and linear search for $r$-NN in high-dimensional space. By integrating an auxiliary data structure into LSH hash tables, we can efficiently estimate the computational cost of LSH-based search for a given query regardless of the data distribution. This means that we are able to choose the appropriate search strategy between LSH-based search and linear search to achieve better performance. Moreover, the integrated data structure is time efficient and fits well with many recent state-of-the-art LSH-based approaches. Our experiments on real-world datasets show that the hybrid search approach outperforms (or is comparable to) both LSH-based search and linear search for a wide range of search radii and data distributions in high-dimensional space.Comment: Accepted as a short paper in EDBT 201