共线点上某些优化问题的快速算法
Faster Algorithms for some Optimization Problems on Collinear Points
摘要：
我们提出了以下三个优化问题的快速算法在$n$共线点，即，第一维度的点。前两个问题是已知的NP-难在更高的维度。1-最大化不相交圆盘的总面积：在这个问题中，目标是最大化以点为中心的不重叠圆盘的总面积。Acharyya，De和Nandy（2017）提出了一个解决这个问题的$O(n^2)$时间算法。我们提出了一个最佳的$Θ(n)$时间算法。2-最小化客户端-服务器覆盖半径之和：将$n$点划分为两个集合，即客户端和服务器。目标是最小化以服务器为中心的磁盘半径之和，使得每个客户端都在某个磁盘中，即，在某个服务器的覆盖范围内Lev-Tov和Peleg（2005）提出了一个求解该问题的$O(n^3)$时间算法。我们提出了一个$O(n^2)$时间算法，从而提高了运行时间的一个因素的$Θ(n)$。3-最小化点区间覆盖的总面积：$n$输入点属于区间$I$。目标是找到覆盖$I$的总面积最小的一组$n$盘，使得每个盘包含至少一个输入点。我们提出了一个算法，解决了这个问题，在$O(n^2)$时间。
英文摘要：
We propose faster algorithms for the following three optimization problems on $n$ collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions. 1- Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an $O(n^2)$-time algorithm for this problem. We present an optimal $Θ(n)$-time algorithm. 2- Minimizing sum of the radii of client-server coverage: The $n$ points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an $O(n^3)$-time algorithm for this problem. We present an $O(n^2)$-time algorithm, thereby improving the running time by a factor of $Θ(n)$. 3- Minimizing total area of point-interval coverage: The $n$ input points belong to an interval $I$. The goal is to find a set of $n$ disks of minimum total area, covering $I$, such that every disk contains at least one input point. We present an algorithm that solves this problem in $O(n^2)$ time.
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