Optimal Point Movement for Covering Circular Regions [article]

Danny Z. Chen and Xuehou Tan and Haitao Wang and Gangshan Wu
2011 arXiv   pre-print
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^2 n)
more » ... algorithm for the decision version and an O(n^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 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(^2 n) time. This result is interesting in its own right.
arXiv:1107.1012v1 fatcat:b472zaxdkncldlszshrfgbn5sq