计算分隔两组线段的最大圆

多边形的圆可分性

Circular Separability of Polygons

两个平面集合是圆可分的，如果存在一个包围其中一个集合的圆，并且其开放的内盘不与另一个集合相交。本文研究了与循环可分性有关的两个问题。提出了一种判定两个多边形是否圆可分的线性时间算法。该算法输出最小的分离圆。第二个问题要求在一些点和/或线约束下，在预处理的凸多边形中包含的最大圆。生成的圆必须包含查询点，并且必须位于由查询线分隔的半平面内。

Efficient Algorithms for Obnoxious Facility Location on a Line Segment or Circle

We study different restricted variations of the obnoxious facility location problem on a plane. The first is the constrained obnoxious facility location on a line segment (COFL-Line) problem. We provide an efficient algorithm for this problem that executes in $O(n ^ 2 \log k + n \log k \log (n^2 + k))$ time. Our result improves on the best known result of $O((nk)^2 \log(nk) + (n + k) \log (nk))$ time obtained by Singireddy and Basappa\cite{singireddy2022dispersing}. We also study the same problem where the facilities must be placed on a given circle (the constrained obnoxious facility location…

Critical Placements of a Square or Circle amidst Trajectories for Junction Detection

Motivated by automated junction recognition in tracking data, we study a problem of placing a square or disc of fixed size in an arrangement of lines or line segments in the plane. We let distances among the intersection points of the lines and line segments with the square or circle define a clustering, and study the complexity of \emph{critical} placements for this clustering. Here critical means that arbitrarily small movements of the placement change the clustering. A parameter $\varepsilon$ defines the granularity of the clustering. Without any assumptions on $\varepsilon$, the critical p…

Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

In this paper we consider the problem of locating $k$ obnoxious facilities (congruent disks of maximum radius) amidst $n$ demand points (existing repulsive facility sites) ordered from left to right in the plane so that none of the existing facility sites are affected (no demand point falls in the interior of the disks). We study this problem in two restricted settings: (i) the obnoxious facilities are constrained to be centered on along a predetermined horizontal line segment $\bar{pq}$, and (ii) the obnoxious facilities are constrained to lie on the boundary arc of a predetermined disk $\cal…

Drawing Graphs on Few Circles and Few Spheres

Given a drawing of a graph, its \emph{visual complexity} is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a different measure for the visual complexity, the \emph{affine cover number}, which is the minimum number of lines (or planes) that together cover a crossing-free straight-line drawing of a graph $G$ in 2D (3D). In this paper, we introduce the \emph{spherical cover number}, which is the minimum number of circ…

平面中分离两点的次二次近似算法

Subquadratic Approximation Algorithms for Separating Two Points with Objects in the Plane

本文提出了一种次二次算法，用于在平面中分离两点，能够以更高效的计算时间生成近似解。