Assigning Weights to Minimize the Covering Radius in the Plane

Given a set P of n points in the plane and a multiset W of k weights with k leq n, we assign a weight in W to a point in P to minimize the maximum weighted distance from the weighted center of P to any point in P. In this paper, we give two algorithms which take O(k^2 n^2 log^4 n) time and O(k^5 n log^4 k + kn log^3 n) time, respectively. For a constant k, the second algorithm takes only O(n log^3 n) time, which is near-linear

How to Cover a Point Set with a V-Shape of Minimum Width

A balanced V-shape is a polygonal region in the plane contained in the union of two crossing equal-width strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are mirror-symmetric with respect to the line xy. The width of a balanced V-shape is the width of the strips. We first present an O(n^2 log n) time algorithm to compute, given a set of n points P, a minimum-width balanced V-shape covering P. We then describe a PTAS for computing a (1+epsilon)-approximation of this V-shape in time O((n/epsilon)log n+(n/epsilon^(3/2))log^2(1/epsilon)). A much simpler constant-factor approximation algorithm is also described.Comment: In Proceedings of the 12th International Symposium on Algorithms and Data Structures (WADS), p.61-72, August 2011, New York, NY, US

On some geometric optimization problems.

An optimization problem is a computational problem in which the objective is to find the best of all possible solutions. A geometric optimization problem is an optimization problem induced by a collection of geometric objects. In this thesis we study two interesting geometric optimization problems. One is the all-farthest-segments problem in which given n points in the plane, we have to report for each point the segment determined by two other points that is farthest from it. The principal motive for studying this problem was to investigate if this problem could be solved with a worst-case time-complexity that is of lower order than O(n 2), which is the time taken by the solution of Duffy et al. (13) for the all-closest version of the same problem. If h be the number of points on the convex hull of the point set, we show how to do this in O(nh + n log n) time. Our solution to this problem has also triggered off research into the hitherto unexplored problem of determining the farthest-segment Voronoi Diagram of a given set of n line segments in the plane, leading to an O(n log n) time solution for the all-farthest-segments problem (12). For the second problem, we have revisited the problem of computing an area-optimal convex polygon stabbing a set of parallel line segments studied earlier by Kumar et al. (30). The primary motive behind this was to inquire if the line of attack used for the parallel-segments version can be extended to the case where the line segments are of arbitrary orientation. We have provided a correctness proof of the algorithm, which was lacking in the above-cited version. Implementation of geometric algorithms are of great help in visualizing the algorithms, we have implemented both the algorithms and trial versions are available at www.davinci.newcs.uwindsor.ca/ ∼asishm.Dept. of Computer Science. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2006 .C438. Source: Masters Abstracts International, Volume: 45-01, page: 0349. Thesis (M.Sc.)--University of Windsor (Canada), 2006

Algorithmic techniques for geometric optimization

Abstract. We review the recent progress in the design of e cient algorithms for various problems in geometric optimization. The emphasis in this survey is on the techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LP-type problems and their e cient solution.