Algorithms for Computing Closest Points for Segments

Given a set $P$ of $n$ points and a set $S$ of $n$ segments in the plane, we consider the problem of computing for each segment of $S$ its closest point in $P$. The previously best algorithm solves the problem in $n^{4/3}2^{O(\log^*n)}$ time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) $\Omega(n^{4/3})$ has also been proved. In this paper, we present an $O(n^{4/3})$ time algorithm and thus solve the problem optimally (under the restricted model). In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in $P$. Our new results improve the previous work.Comment: Accepted to STACS 202

Approximate Euclidean shortest paths in polygonal domains

Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $\epsilon$. We utilize $\Omega$ to compute a $(1+\epsilon)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}})))$ time. Here, $\epsilon$ is a user parameter, and $\delta$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+\epsilon)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.Comment: a few updates; accepted to ISAAC 201

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

Computing largest circles separating two sets of segments

A circle $C$ separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect $\Omega(n^2)$ times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on Computational Geometry, 199

Covering the Boundary of a Simple Polygon with Geodesic Unit Disks

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm