Bernd Gärtner

A Novel Type of Skeleton for Polygons

A new internal structure for simple polygons, the straight skeleton, is introduced and discussed. It is composed of pieces of angular bisectores which partition the interior of a given n-gon P in a tree-like fashion into n monotone polygons. Its straight-line structure and its lower combinatorial complexity may make the straight skeleton preferable to the widely used medial axis of a polygon. As a seemingly unrelated application, the straight skeleton provides a canonical way of constructing a polygonal roof above a general layout of ground walls.

Fast and Robust Smallest Enclosing Balls

I describe a C++ program for computing the smallest enclosing ball of a point set in d-dimensional space, using floating-point arithmetic only. The program is very fast for d ? 20, robust and simple (about 300 lines of code, excluding prototype definitions). Its new features are a pivoting approach resembling the simplex method for linear programming, and a robust update scheme for intermediate solutions. The program with complete documentation following the literate programming paradigm [3] is available on the Web.

Fast Smallest-Enclosing-Ball Computation in High Dimensions

We develop a simple combinatorial algorithm for computing the smallest enclosing ball of a set of points in high dimensional Euclidean space. The resulting code is in most cases faster (sometimes significantly) than recent dedicated methods that only deliver approximate results, and it beats off-the-shelf solutions, based e.g. on quadratic programming solvers. The algorithm resembles the simplex algorithm for linear programming; it comes with a Bland-type rule to avoid cycling in presence of degeneracies and it typically requires very few iterations. We provide a fast and robust floating-point implementation whose efficiency is based on a new dynamic data structure for maintaining intermediate solutions.

The smallest enclosing ball of balls: combinatorial structure and algorithms

We develop algorithms for computing the smallest enclosing ball of a set of n balls in d-dimensional space. Unlike previous methods, we explicitly address small cases (n= d+1), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An exact implementation (along with a fast For d=3, a set of 1,000,000 balls is processed in less than two seconds on a modern PC. and robust floating-point version) is available as part of the CGAL library.See http://www.cgal.org.Our algorithms are based on novel insights into the combinatorial structure of the problem. As it turns out, results for smallest enclosing balls of points do not extend as one might expect. For example, we show that Welzls randomized linear-time algorithm for computing the ball spanned by a set of points fails to work for balls. Consequently, David Whites adaptation of the method to the ball case---as the only available implementation so far it is mentioned in many link collections---is incorrect and may crash or, in the better case, produce wrong balls.In solving the small cases we may assume that the ball centers are affinely independent; in this case, the problem is surprisingly well-behaved: via a geometric transformation and suitable generalization, it fits into the combinatorial model of unique sink orientations whose rich structure has recently received considerable attention. One consequence is that Welzls algorithm does work for small instances; moreover, there is a wide variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimension.As a by-product, we show that the problem of finding the smallest enclosing ball of balls is computationally equivalent to the problem of finding the minimum-norm point in the convex hull of a set of balls.

Linear Programming - Randomization and Abstract Frameworks

Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial’ algorithm which solves a linear program over d variables and n constraints with expected O(d2n+eO(√d log d)) arithmetic operations. The bound relies on two algorithms by Clarkson, and the subexponential algorithms due to Kalai, and to Matousek, Sharir & Welzl.

Coresets for polytope distance

Following recent work of Clarkson, we translate the coreset framework to the problems of finding the point closest to the origin inside a polytope, finding the shortest distance between two polytopes, Perceptrons, and soft- as well as hard-margin Support Vector Machines (SVM). We prove asymptotically matching upper and lower bounds on the size of coresets, stating that µ-coresets of size (1+o(1)) E*/µ do always exist as µ-0, and that this is best possible. The crucial quantity E* is what we call the excentricity of a polytope, or a pair of polytopes. Additionally, we prove linear convergence speed of Gilberts algorithm, one of the earliest known approximation algorithms for polytope distance, and generalize both the algorithm and the proof to the two polytope case. Interestingly, our coreset bounds also imply that we can for the first time prove matching upper and lower bounds for the sparsity of Perceptron and SVM solutions.

Computing Largest Common Point Sets under Approximate Congruence

The problem of computing a largest common point set (LCP) between two point sets under Ɛ-congruence with the bottleneck matching metric has recently been a subject of extensive study. Although polynomial time solutions are known for the planar case and for restricted sets of transformations and metrics (like translations and the Hausdorff-metric under L∞-norm), no complexity results are formally known for the general problem. In this paper we give polynomial time algorithms for this problem under different classes of transformations and metrics for any fixed dimension, and establish NP-hardness for unbounded dimensions. Any solution to this (or related) problem, especially in higher dimensions, is generally believed to involve implementation difficulties because they rely on the computation of intersections between algebraic surfaces. We show that (contrary to intuitive expectations) this problem can be solved under a rational arithmetic model in a straightforward manner if the set of transformations is extended to general affine transformations under the L∞-norm (difficulty of this problem is generally expected to be in the order: translations < rotation < isometry < more general). To the best of our knowledge this is also the first paper which deals with the LCP-problem under such a general class of transformations.

Randomized Simplex Algorithms on Klee-Minty Cubes

The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature.At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.

A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization

Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms.Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma ) which holds in a general framework, yet has not been stated explicitly in the literature.In the more restricted but still general setting of LP-type problems , we prove tail estimates for the sampling lemma, giving Chernoff-type bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple pricing , a heuristic used in the simplex method for linear programming in order to reduce a large problem to few small ones. This follows from our analysis of a reduction scheme for general LP-type problems, which can be considered as a simplification of an algorithm due to Clarkson. The simplified version needs less random resources and allows a Chernoff-type tail estimate.

An efficient, exact, and generic quadratic programming solver for geometric optimization

We present a solver for quadratic programming problems, which is tuned for applications in computational geometry. The solver implements a generalization of the simplex method to quadratic programs. Unlike existing solvers, it is e cient if the problem is dense and has few variables or few constraints. The range of applications covers well-known problems like smallest enclosing ball, or polytope distance, but also linear programming problems like smallest enclosing annulus. We provide an exact implementation with only little overhead compared to pure oating-point code. Moreover, unlike all methods for these problems that were suggested (and implemented) before in computational geometry, the runtime in practice is not exponential in the dimension of the problem, which for example allows to compute smallest enclosing balls in dimensions up to 300 (beyond that, the exact arithmetic becomes the limiting factor). The solver follows the generic programming paradigm, and it will become part of the European computational geometry algorithms library Cgal.