Fitting Voronoi Diagrams to Planar Tesselations

Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ "fits" \ $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.Comment: 14 pages, 8 figures, 1 table. Presented at IWOCA 2013 (Int. Workshop on Combinatorial Algorithms), Rouen, France, July 201

Largest Empty Circle Centered on a Query Line

The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set $P$ of $n$ points in $\mathbb{R}^2$ and devoid of points from $P$. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess $P$ so that when given a query line $Q$, we can quickly compute the largest empty circle centered at some point on $Q$ and within the convex hull of $P$. We present solutions for two special cases and the general case; all our queries run in $O(\log n)$ time. We restrict the query line to be horizontal in the first special case, which we preprocess in $O(n \alpha(n) \log n)$ time and space, where $\alpha(n)$ is the slow growing inverse of the Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes $O(n \alpha(n)^{O(\alpha(n))} \log n)$ time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in $O(n^3 \log n)$ and $O(n^3)$ respectively.Comment: 18 pages, 13 figure

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

Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time

In this paper, we present a Θ(n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.published_or_final_versio

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology