Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

Vishwanath R. Singireddy, Manjanna Basappa
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引用次数: 2

Abstract

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 $\cal C$. An $(1-\epsilon)$-approximation algorithm was given recently to solve the constrained problem in (i) in time $O((n+k)\log{\frac{||pq||}{2(k-1)\epsilon}})$, where $\epsilon>0$ \cite{Sing2021}. Here, for the problem in (i), we first propose an exact polynomial-time algorithm based on a binary search on all candidate radii computed explicitly. This algorithm runs in $O((nk)^2\log{(nk)}+(n+k)\log{(nk)})$ time. We then show that using the parametric search technique of Megiddo \cite{MG1983}; we can solve the problem exactly in $O((n+k)^2)$ time, which is faster than the latter. Continuing further, using the improved parametric technique we give an $O(n\log^2 n)$-time algorithm for $k=2$. We finally show that the above $(1-\epsilon)$-approximation algorithm of \cite{Sing2021} can be easily adapted to solve the circular constrained problem of (ii) with an extra multiplicative factor of $n$ in the running time.
平面段和圆上的分散设施
在本文中,我们考虑在平面上从左到右排列的$n$需求点(现有的排斥设施地点)中定位$k$讨厌设施(最大半径的同余磁盘)的问题,以使现有的设施地点不受影响(没有需求点落在磁盘的内部)。我们在两种限制条件下研究了这个问题:(i)将讨厌设施限制在沿预定水平线段$\bar{pq}$的中心,以及(ii)将讨厌设施限制在预定圆盘的边界弧$\cal C$上。最近给出了一个$(1-\epsilon)$ -逼近算法来求解(i)中的约束问题,时间为$O((n+k)\log{\frac{||pq||}{2(k-1)\epsilon}})$,其中$\epsilon>0$\cite{Sing2021}。在这里,对于(i)中的问题,我们首先提出了一种精确多项式时间算法,该算法基于对明确计算的所有候选半径的二分搜索。该算法运行时间为$O((nk)^2\log{(nk)}+(n+k)\log{(nk)})$。然后我们用参数搜索技术证明了Megiddo \cite{MG1983};我们可以精确地在$O((n+k)^2)$时间内解决这个问题,这比后者要快。进一步,利用改进的参数化技术,我们给出了$k=2$的$O(n\log^2 n)$时间算法。我们最后表明,上述\cite{Sing2021}的$(1-\epsilon)$ -近似算法可以很容易地适用于解决(ii)的圆形约束问题,在运行时间上增加了$n$的乘因子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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