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.
KeywordsComputational geometry Algorithms and data structures Circular region coverage Barrier coverage Mobile sensors Dynamic maximum matching Circular convex bipartite graph
- 3.Bremner, D., Chan, T.M., Demaine, E.D., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Taslakian, P.: Necklaces, convolutions, and X + Y. In: Proc. of the 14th Conference on Annual European Symposium on Algorithms, pp. 160–171 (2006) Google Scholar
- 4.Brodal, G., Georgiadis, L., Hansen, K.A., Katriel, I.: Dynamic matchings in convex bipartite graphs. In: Proc. of the 32nd International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 4708, pp. 406–417. Springer, Berlin (2007) Google Scholar
- 7.Chen, A., Kumar, S., Lai, T.: Designing localized algorithms for barrier coverage. In: Proc. of the 13th Annual ACM International Conference on Mobile Computing and Networking, pp. 63–73 (2007) Google Scholar
- 15.Hu, S.: ‘Virtual Fence’ along border to be delayed. Washington Post, February 28, 2008 Google Scholar