Optimal Point Movement for Covering Circular Regions

Danny Z. Chen, Xuehou Tan, Haitao Wang, Gangshan Wu
2013 Algorithmica  
Given n points in a circular region C in the plane, we study the problems of moving the n points to the boundary of G 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. These 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 the optimization version. For the min-max problem, we present an
more » ... 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. In addition, a by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; Algorithmica (2015) 72:379-399 our algorithm can handle each vertex insertion or deletion on the graph in O(log 2 n) time. This result may be interesting in its own right.
doi:10.1007/s00453-013-9857-1 fatcat:wkgw4s3wkbbkfhmeplaslfpftm