Economical Delone Sets for Approximating Convex Bodies

Convex bodies are ubiquitous in computational geometry and optimization theory. The high combinatorial complexity of multidimensional convex polytopes has motivated the development of algorithms and data structures for approximate representations. This paper demonstrates an intriguing connection between convex approximation and the classical concept of Delone sets from the theory of metric spaces. It shows that with the help of a classical structure from convexity theory, called a Macbeath region, it is possible to construct an epsilon-approximation of any convex body as the union of O(1/epsilon^{(d-1)/2}) ellipsoids, where the center points of these ellipsoids form a Delone set in the Hilbert metric associated with the convex body. Furthermore, a hierarchy of such approximations yields a data structure that answers epsilon-approximate polytope membership queries in O(log (1/epsilon)) time. This matches the best asymptotic results for this problem, by a data structure that both is simpler and arguably more elegant

Enumerating a subset of the integer points inside a Minkowski sum

AbstractSparse elimination exploits the structure of algebraic equations in order to obtain tighter bounds on the number of roots and better complexity in numerically approximating them. The model of sparsity is of combinatorial nature, thus leading to certain problems in general-dimensional convex geometry. This work addresses one such problem, namely the computation of a certain subset of integer points in the interior of integer convex polytopes. These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm. Complexity bounds for our algorithm are derived under certain hypotheses, in terms of output-size and the sparsity parameters. A public domain implementation is described and its performance studied. Linear optimization lies at the inner loop of the algorithm, hence we analyze the structure of the linear programs and compare different implementations

Fixed-Dimensional Linear Programming Queries Made Easy

We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized data structure with O(n) space and O(n 1\Gamma1=bd=2c 2 O(log n) ) query time for linear programming on n halfspaces (d ? 3). The second result is a simpler proof of the following: a sequence of q linear programming queries on n halfspaces can be answered in O(n log q) time, if q n ff d for a certain constant ff d ? 0. Unlike previous methods, our algorithms do not require parametric searching. 1 Introduction One of the major discoveries in computational geometry is that fixed-dimensional linear programming can be solved in linear time [Meg84]. It was observed that the introduction of randomization leads to considerably simpler solutions [Sei91, Cla95]. The goal of this paper is..

Approximate Polytope Membership Queries

International audienceIn the polytope membership problem, a convex polytope K in R d is given, and the objective is to preprocess K into a data structure so that, given any query point q ∈ R d , it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting. Given an approximation parameter ε, the query can be answered either way if the distance from q to K's boundary is at most ε times K's diameter. We assume that the dimension d is fixed, and K is presented as the intersection of n halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley (1974) and Bentley et al. (1982). The former is optimal in the worst-case with respect to space, and the latter is optimal with respect to query time. We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant α ≥ 4, this data structure achieves query time roughly O 1/ε (d−1)/α and space roughly O 1/ε (d−1)(1−Ω(log α)/α). We do not know whether this space bound is tight, but our third result shows that there is a convex body such that our algorithm achieves a space of at least Ω 1/ε (d−1)(1−O(√ α)/α. Our fourth result shows that it is possible to reduce approximate Euclidean nearest neighbor searching to approximate polytope membership queries. Combined with the above results, this provides significant improvements to the best known space-time trade-offs for approximate nearest neighbor searching in R d. For example, we show that it is possible to achieve a query time of roughly O(log n + 1/ε d/4) with space roughly O(n/ε d/4), thus reducing by half the exponent in the space bound

GEOMETRIC OPTIMIZATION IN SOME PROXIMITY AND BIOINFORMATICS PROBLEMS

The theme of this dissertation is geometric optimization and its applications. We study geometric proximity problems and several bioinformatics problems with a geometric content, requiring the use of geometric optimization tools. We have investigated the following type of proximity problems. Given a point set in a plane with n distinct points, for each point in the set find a pair of points from the remaining points in the set such that the three points either maximize or minimize some geometric measure defined on these. The measures include (a) sum and product; (b) difference; (c) line–distance; (d) triangle area; (e) triangle perimeter; (f) circumcircle–radius; and (g) triangle–distance in three dimensions. We have also studied the application of a linear time incremental geometric algorithm to test the linear separability of a set of blue points from a set of red points, in two and three–dimensional Euclidean spaces. We have used this geometric separability tool on 4 different gene expression data–sets, enumerating gene–pairs and gene–triplets that are linearly separable. Pushing on further, we have exploited this novel tool to identify some bio–marker genes for a classifier. The gene selection method proposed in the dissertation exhibits good classification accuracy as compared to other known feature (or gene) selection methods such as t–values, FCS (Fisher Criterion Score) and SAM (Significance Analysis of Microarrays). Continuing this line of investigation further, we have also designed an efficient algorithm to find the minimum number of outliers when the red and blue point sets are not fully linearly separable. We have also explored the applicability of geometric optimization techniques to the problem of protein structure similarity. We have come up with two new algorithms, EDAlignres and EDAlignsse, for pairwise protein structure alignment. EDAlignres identifies the best structural alignment of two equal length proteins by refining the correspondence obtained from eigendecomposition and to maximize the similarity measure for the refined correspondence. EDAlignsse, on the other hand, does not require the input proteins to be of equal length. These have been fully implemented and tested against well-established protein alignment program

Adaptive Sampling for Geometric Approximation

Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells