覆盖圆形区域的最优点移动
Optimal Point Movement for Covering Circular Regions
摘要：
给定平面上圆形区域$C$中的$n$点，研究了将$n$点移动到其边界形成正则$n$ -边形，使得点所经过的欧氏距离的最大值（min-max）或和（min-sum）最小化的问题.这些问题具有应用，例如，在移动传感器屏障覆盖的无线传感器网络。最小-最大问题还有两个版本：决策版本和优化版本。对于最小-最大问题，我们提出了一个$O(n\log^2 n)$时间算法的决策版本和$O(n\log^3 n)$时间算法的优化版本。对于这两个问题版本，以前最好的算法分别需要$O(n^{3.5})$时间和$O(n^{3.5}\log n)$时间。对于最小和问题，我们表明，一个特殊的情况下，所有的点最初躺在边界上的圆形区域可以解决在$O(n^2)$时间，提高以前的$O(n^4)$时间解决方案。对于一般的最小和问题，我们提出了一个3-近似$O(n^2)$时间算法，改进了以前的$(1+π)$ -近似$O(n^2)$时间算法.我们的技术的一个副产品是一个算法，用于动态保持最大匹配的圆形凸二分图，我们的算法可以处理每个顶点插入或删除的图形在$O(\log^2 n)$时间。这个结果本身就很有趣。
英文摘要：
Given $n$ points in a circular region $C$ in the plane, we study the problems of moving the $n$ points to its boundary to form a regular $n$-gon such that the maximum (min-max) or the sum (min-sum) of the Euclidean distances traveled by the points is minimized. The problems have applications, e.g., in mobile sensor barrier coverage of wireless sensor networks. The min-max problem further has two versions: the decision version and optimization version. For the min-max problem, we present an $O(n\log^2 n)$ time algorithm for the decision version and an $O(n\log^3 n)$ time algorithm for the optimization version. The previously best algorithms for the two problem versions take $O(n^{3.5})$ time and $O(n^{3.5}\log n)$ time, respectively. For the min-sum problem, we show that a special case with all points initially lying on the boundary of the circular region can be solved in $O(n^2)$ time, improving a previous $O(n^4)$ time solution. For the general min-sum problem, we present a 3-approximation $O(n^2)$ time algorithm, improving the previous $(1+π)$-approximation $O(n^2)$ time algorithm. A by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm can handle each vertex insertion or deletion on the graph in $O(\log^2 n)$ time. This result is interesting in its own right.
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