Yam Ki Cheung

Minimum Separating Circle for Bichromatic Points in the Plane

Consider two point sets in the plane, a red set of size n, and a blue set of size m. In this paper we show how to find the minimum separating circle, which is the smallest circle that contains all points of the red set and as few points as possible of the blue set in its interior. If multiple minimum separating circles exist our algorithm finds all of them. We also give an exact solution for finding the largest separating circle that contains all points of the red set and as few points as possible of the blue set in its interior. Our solutions make use of the farthest neighbor Voronoi Diagram of point sites.

Parallel Optimal Weighted Links

In this paper we consider parallel algorithms for computing an optimal link among weighted regions in the plane. The problem arises in several areas, including radiation therapy, geological exploration and environmental engineering. We present a CREW PRAM parallel algorithm and a coarse-grain parallel computer algorithm for this problem. For a weighted subdivision with n vertices, the work of the parallel algorithms we propose is only an O (logn ) factor more than that of their optimal sequential counterparts. We further adapt an algorithm for minimizing sum of linear fractionals, that has inherent parallelism, to solve in parallel the global optimization problems associated with our solution for the weighted region optimal link problem.

Kinetic red-blue minimum separating circle

In this paper, we study a kinetic version of the red-blue minimum separating circle problem, in which some points move with constant speed along straight line trajectories. We want to find the locus of the minimum separating circle over a period of time. We first consider two degenerate cases of this problem. In the first one (P1), we study the minimum separating circle problem with only one mobile blue point, and in the second one (P2), we study the minimum separating circle problem with only one mobile red point. Then, we give a solution for the general case (P3), in which multiple points are mobile.

Visiting a sequence of points with a bevel-tip needle

Many surgical procedures could benefit from guiding a bevel-tip needle along circular arcs to multiple treatment points in a patient. At each treatment point, the needle can inject a radioactive pellet into a cancerous region or extract a tissue sample. Our main result is an algorithm to steer a bevel-tip needle through a sequence of treatment points in the plane while minimizing the number of times that the needle must be reoriented. This algorithm is related to [6] and takes quadratic time when consecutive points in the sequence are sufficiently separated. We can also guide a needle through an arbitrary sequence of points in the plane by accounting for a lack of optimal substructure.

A New Modeling for Finding Optimal Weighted Distances

We prove that each optimization problem associated with finding an optimal straight line distance between two regions of a weighted planar subdivision can be restated as a two-dimensional sum of linear fractionals problem over an arc of the unit circle. Compared to previous results, that involved more general functions over two dimensional domains, our solution has potential for order of magnitude speedups. The problem has a few bio-medical applications, including optimal treatment planning in intensity modulated radiation therapy and brachytherapy.

Fréchet Distance Problems in Weighted Regions

We discuss two versions of the Frechet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases we give algorithms for finding a (1 + ?)-factor approximation of the Frechet distance between two polygonal curves. We also consider the Frechet distance between two polygonal curves among polyhedral obstacles in

Line Facility Location in Weighted Regions

\mathcal{R}^3

Line facility location in weighted regions

(1/ ? weighted region problem) and present a (1 + ?)-factor approximation algorithm.

FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS

In this paper, we present approximation algorithms for the line facility location problem in weighted regions: Given lfixed points in a 2-dimensional weighted subdivision of the plane, with nvertices, find a line Lsuch that the sum of the weighted distances from the fixed points to Lis minimized. The weighted region setup is a more realistic model for many facility location problems that arise in practical applications. Our algorithms exploit an interesting property of the problem, that could possibly be used for solving other problems in weighted regions.

Line Segment Facility Location in Weighted Subdivisions

In this paper, we present approximation algorithms for solving the line facility location problem in weighted regions. The weighted region setup is a more realistic model for many facility location problems that arise in practical applications. Our algorithms exploit an interesting property of the problem, that could possibly be used for solving other problems in weighted regions.