Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums

Approximation problems involving a single convex body in R^d have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions d 0, we show how to independently preprocess two polytopes A,B subset R^d into data structures of size O(1/epsilon^{(d-1)/2}) such that we can answer in polylogarithmic time whether A and B intersect approximately. More generally, we can answer this for the images of A and B under affine transformations. Next, we show how to epsilon-approximate the Minkowski sum of two given polytopes defined as the intersection of n halfspaces in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to epsilon-approximate the width of a set of n points in O(n log(1/epsilon) + 1/epsilon^{(d-1)/2 + alpha}) time, for any constant alpha > 0, a major improvement over the previous bound of roughly O(n + 1/epsilon^{d-1}) time

Asymptotic Geometric Analysis

The workshop was dedicated to new developments in Asymptotic Geometric Analysis, the study of high-dimensional geometric objects by analytic and probabilistic means, and its interplay with other fields such as functional analysis, convex geometry, probability and graph theory. A central aspect concerned volume inequalities for sections and projections of high-dimensional convex sets

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Compaction and Separation Algorithms for Non-Convex Polygons and Their Applications

Given a two dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied "forces." We apply compaction to improve the material utilization of an already tightly packed layout. Compaction can be modeled as a motion of the polygons that reduces the value of some functional on their positions. Optimal compaction, planning a motion that reaches a layout that has the global minimum functional value among all reachable layouts, is shown to be NP-complete under certain assumptions. We first present a compaction algorithm based on existing physical simulation approaches. This algorithm uses a new velocity-based optimization model. Our experimental results reveal the limitation of physical simulation: even though our new model improves the running time of our algorithm over previous simulation algorithms, the algorithm still can not compact typical layouts of one hundred or more polygons in a reasonable amount of time. The essential difficulty of physical based models is that they can only generate velocities for the polygons, and the final positions must be generated by numerical integration. We present a new position-based optimization model that allows us to calculate directly new polygon positions via linear programming that are at a local minimum of the objective. The new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods. We also consider the problem of separating overlapping polygons using a minimal amount of motion and show it to be NP-complete. Although this separation problem looks quite different from the compaction problem, our new model also yields an efficient algorithm to solve it. The compaction/separation algorithms have been applied to marker making: the task of packing polygonal pieces on a sheet of cloth of fixed width so that total length is minimized. The compaction algorithm has improved cloth utilization of human generated pants markers. The separation algorithm together with a database of human-generated markers can be used for automatic generation of markers that approach human performance.Engineering and Applied Science

Set-valued Data: Regression, Design and Outliers

The focus of this dissertation is to study set‐valued data from three aspects, namely regression, optimal design and outlier identification. This dissertation consists of three peer‐reviewed published articles, each of them addressing one aspect. Their titles and abstracts are listed below: 1. Local regression smoothers with set‐valued outcome data: This paper proposes a method to conduct local linear regression smoothing in the presence of set‐valued outcome data. The proposed estimator is shown to be consistent, and its mean squared error and asymptotic distribution are derived. A method to build error tubes around the estimator is provided, and a small Monte Carlo exercise is conducted to confirm the good finite sample properties of the estimator. The usefulness of the method is illustrated on a novel dataset from a clinical trial to assess the effect of certain genes’ expressions on different lung cancer treatments outcomes. 2. Optimal design for multivariate multiple linear regression with set‐identified response: We consider the partially identified regression model with set‐identified responses, where the estimator is the set of the least square estimators obtained for all possible choices of points sampled from set‐identified observations. We address the issue of determining the optimal design for this case and show that, for objective functions mimicking those for several classical optimal designs, their set‐identified analogues coincide with the optimal designs for point‐identified real‐valued responses. 3. Depth and outliers for samples of sets and random sets distributions: We suggest several constructions suitable to define the depth of set‐valued observations with respect to a sample of convex sets or with respect to the distribution of a random closed convex set. With the concept of a depth, it is possible to determine if a given convex set should be regarded an outlier with respect to a sample of convex closed sets. Some of our constructions are motivated by the known concepts of half‐space depth and band depth for function‐valued data. A novel construction derives the depth from a family of non‐linear expectations of random sets. Furthermore, we address the role of positions of sets for evaluation of their depth. Two case studies concern interval regression for Greek wine data and detection of outliers in a sample of particles