论文标题
平面细分市场的分散设施，并在排斥力中圆圈
Dispersing Facilities on Planar Segment and Circle Amidst Repulsion
论文作者
论文摘要
在本文中，我们考虑了在$ n $需求点（现有的排斥设施站点）中定位$ k $令人讨厌的设施（最大半径的一致磁盘）的问题。我们在两个受限制的设置中研究了这个问题：（i）令人讨厌的设施被限制在预定的水平线段$ \ bar {pq} $上，以及（ii）令人讨厌的设施被限制在预定的磁盘$ \ c $的边界上。最近给出了一个$（1-ε）$ - 近似算法，以解决（i）时间$ o（（（n+k）\ log {\ frac {\ frac {|| pq ||} {|| pq || 2（k-1）{k-1）ε}}} $，其中$ε> 0 $ε> 0 $ \ cite {sing2021}。在这里，对于（i）中的问题，我们首先根据对所有计算的所有候选半径进行了基于二进制搜索的精确多项式算法。该算法在$ o（（nk）^2 \ log {（nk）}+（n+k）\ log {（nk）}）$ time中运行。然后，我们证明使用Megiddo \ cite {mg1983}的参数搜索技术;我们可以在$ o（（n+k）^2）$时间中准确解决问题，这比后者快。继续进一步，使用改进的参数技术，我们给出了$ o（n \ log^2 n）$ - $ k = 2 $的时间算法。我们最终表明，可以很容易地适应上述$（1-ε）$ - 近似算法\ cite {sing2021}，以解决（ii）的圆形约束问题（ii），并且在运行时间内额外的乘法系数为$ n $。
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 $(1-ε)$-approximation algorithm was given recently to solve the constrained problem in (i) in time $O((n+k)\log{\frac{||pq||}{2(k-1)ε}})$, where $ε>0$ \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 $(1-ε)$-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.
