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 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 $(1+\pi)$-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.