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Algorithmica

, Volume 72, Issue 2, pp 379–399 | Cite as

Optimal Point Movement for Covering Circular Regions

  • Danny Z. Chen
  • Xuehou Tan
  • Haitao Wang
  • Gangshan Wu
Article

Abstract

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.

Keywords

Computational geometry Algorithms and data structures Circular region coverage Barrier coverage Mobile sensors Dynamic maximum matching Circular convex bipartite graph 

References

  1. 1.
    Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: Wireless sensor networks: a survey. Comput. Netw. 38(4), 393–422 (2002) CrossRefGoogle Scholar
  2. 2.
    Bhattacharya, B., Burmester, B., Hu, Y., Kranakis, E., Shi, Q., Wiese, A.: Optimal movement of mobile sensors for barrier coverage of a planar region. Theor. Comput. Sci. 410(52), 5515–5528 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 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. 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
  5. 5.
    Buss, S., Yianilos, P.: Linear and O(nlogn) time minimum-cost matching algorithms for quasi-convex tours. SIAM J. Comput. 27(1), 170–201 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chang, M.S., Tang, C.Y., Lee, R.C.T.: Solving the Euclidean bottleneck matching problem by k-relative neighborhood graphs. Algorithmica 8, 177–194 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 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
  8. 8.
    Chen, D.Z., Wang, C., Wang, H.: Representing a functional curve by curves with fewer peaks. Discrete Comput. Geom. 46(2), 334–360 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34(1), 200–208 (1987) CrossRefGoogle Scholar
  10. 10.
    Cole, R., Salowe, J., Steiger, W., Szemerédi, E.: An optimal-time algorithm for slope selection. SIAM J. Comput. 18(4), 792–810 (1989) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dillencourt, M.B., Mount, D.M., Netanyahu, N.S.: A randomized algorithm for slope selection. Int. J. Comput. Geom. Appl. 2, 1–27 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Efrat, A., Katz, M.J.: Computing Euclidean bottleneck matchings in higher dimensions. Inf. Process. Lett. 75, 169–174 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Efrat, A., Itai, A., Katz, M.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gabow, H., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30, 209–221 (1985) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hu, S.: ‘Virtual Fence’ along border to be delayed. Washington Post, February 28, 2008 Google Scholar
  16. 16.
    Katz, M., Sharir, M.: Optimal slope selection via expanders. Inf. Process. Lett. 47(3), 115–122 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kumar, S., Lai, T., Arora, A.: Barrier coverage with wireless sensors. Wirel. Netw. 13(6), 817–834 (2007) CrossRefGoogle Scholar
  18. 18.
    Liang, Y.D., Blum, N.: Circular convex bipartite graphs: maximum matching and Hamiltonian circuits. Inf. Process. Lett. 56, 215–219 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lipski, W. Jr., Preparata, F.P.: Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems. Acta Inform. 15(4), 329–346 (1981) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Matoušek, J.: Randomized optimal algorithm for slope selection. Inf. Process. Lett. 39, 183–187 (1991) CrossRefzbMATHGoogle Scholar
  21. 21.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Steiner, G., Yeomans, J.: A linear time algorithm for maximum matchings in convex, bipartite graphs. Comput. Math. Appl. 31(2), 91–96 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Tan, X., Wu, G.: New algorithms for barrier coverage with mobile sensors. In: Proc. of the 4th International Workshop on Frontiers in Algorithmics. Lecture Notes in Computer Science, vol. 6213, pp. 327–338. Springer, Berlin (2010) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Danny Z. Chen
    • 1
  • Xuehou Tan
    • 2
  • Haitao Wang
    • 3
  • Gangshan Wu
    • 4
  1. 1.Department of Computer Science and EngineeringUniversity of Notre DameNotre DameUSA
  2. 2.Tokai UniversityHiratsukaJapan
  3. 3.Department of Computer ScienceUtah State UniversityLoganUSA
  4. 4.State Key Lab for Novel Software TechnologyNanjing UniversityNanjingChina

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