Hervé Brönnimann

Almost optimal set covers in finite VC-dimension

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous Θ(log⋎X⋎) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.

GPS-Free node localization in mobile wireless sensor networks

An important problem in mobile ad-hoc wireless sensor networks is the localization of individual nodes, i.e., each nodes awareness of its position relative to the network. In this paper, we introduce a variant of this problem (directional localization) where each node must be aware of both its position and orientation relative to the network. This variant is especially relevant for the applications in which mobile nodes in a sensor network are required to move in a collaborative manner. Using global positioning systems for localization in large scale sensor networks is not cost effective and may be impractical in enclosed spaces. On the other hand, a set of pre-existing anchors with globally known positions may not always be available. To address these issues, in this work we propose an algorithm for directional node localization based on relative motion of neighboring nodes in an ad-hoc sensor network without an infrastructure of global positioning systems (GPS), anchor points, or even mobile seeds with known locations. Through simulation studies, we demonstrate that our algorithm scales well for large numbers of nodes and provides convergent localization over time, even with errors introduced by motion actuators and distance measurements. Furthermore, based on our localization algorithm, we introduce mechanisms to preserve network formation during directed mobility in mobile sensor networks. Our simulations confirm that, in a number of realistic scenarios, our algorithm provides for a mobile sensor network that is stable over time irrespective of speed, while using only constant storage per neighbor.

Minimum-cost coverage of point sets by disks

We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t_j)</i> and radii <i>(r_j)</i> that cover a given set of demand points <i>Y∈R</i>² at the smallest possible cost. We consider cost functions of the form Ε<i>_jf(r_j)</i>, where <i>f(r)=r</i>^α is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers <i>t_j</i>, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points <i>t_j</i> on a given line in order to cover demand points <i>Y∈R</i>²; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R²</i> and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

Towards in-place geometric algorithms and data structures

For many geometric problems, there are efficient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are efficient data structures that need no extra space at all other than an array holding a permutation of the input. In this paper, we obtain the first such space-economical solutions for a number of fundamental problems, including three-dimensional convex hulls, two-dimensional Delaunay triangulations, fixed-dimensional range queries, and fixed-dimensional nearest neighbor queries.

The design of the Boost interval arithmetic library

We present the design of the Boost interval arithmetic library, a C++ library designed to handle mathematical intervals efficiently and in a generic way. Interval computations are an essential tool for reliable computing. Increasingly a number of mathematical proofs have relied on global optimization problems solved using branch-and-bound algorithms with interval computations; it is therefore extremely important to have a mathematically correct implementation of interval arithmetic. Various implementations exist with diverse semantics. Our design is unique in that it uses policies to specify three independent variable behaviors: rounding, checking, and comparisons. As a result, with the proper policies, our interval library is able to emulate almost any of the specialized libraries available for interval arithmetic, without any loss of performance nor sacrificing the ease of use. This library is openly available at www.boost.org.

Sign determination in residue number systems

Sign determination is a fundamental problem in algebraic as well as geometric computing. It is the critical operation when using real algebraic numbers and exact geometric predicates. We propose an exact and efficient method that determines the sign of a multivariate polynomial expression with rational coefficients. Exactness is achieved by using modular computation. Although this usually requires some multiprecision computation, our novel techniques of recursive relaxation of the moduli and their variants enable us to carry out sign determination and comparisons by using only single precision. Moreover, to exploit modern day hardware, we exclusively rely on floating point arithmetic, which leads us to a hybrid symbolic-numeric approach to exact arithmetic. We show how our method can be used to generate robust and efficient implementations of real algebraic and geometric algorithms including Sturm sequences, algebraic representation of points and curves, convex hull and Voronoi diagram computations and solid modeling. This method is highly parallelizable, easy to implement, and compares favorably with known multiprecision methods from a practical complexity point of view. We substantiate these claims by experimental results and comparisons to other existing approaches.

Product range spaces, sensitive sampling, and derandomization

We introduce the concept of a sensitive /spl epsi/-approximation, and use it to derive a more efficient algorithm for computing /spl epsi/-nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(nlog/sup 3/ n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space.<<ETX>>

Optimal slope selection via cuttings

Abstract We give an optimal deterministic O(n log n)-time algorithm for slope selection. The algorithm borrows from the optimal solution given in (Cole et al., 1989) but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n)-time algorithm of Chazelle et al. (1993) with the help of additional approximation tools.

Efficient exact evaluation of signs of determinants

This paper presents a theoretical and experimental study on two different methods to evaluate the sign of a determinant with integer entries. The first one is a method based on the Gram—Schmidt orthogonalization process which has been proposed by Clarkson [Cl]. We review his algorithm and propose a variant of his method, for which we give a complete analysis. The second method is an extension to n × n determinants of the ABDPY method [ABD+2] which works only for 2 × 2 and 3 × 3 determinants. Both methods compute the sign of an n× n determinant whose entries are integers on b bits, by using exact arithmetic on only b +O(n) bits. Furthermore, both methods are adaptive, dealing quickly with easy cases and resorting to full-length computation only for null determinants.

Space-efficient planar convex hull algorithms

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.