Optimal Point Movement for Covering Circular Regions

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 O(nlog² n) time algorithm for the decision version and an O(nlog³ 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 ^3.5logn) 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; our algorithm can handle each vertex insertion or deletion on the graph in O(log² n) time. This result may be interesting in its own right.