Journal of information and communication convergence engineering

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
권호 표지
DOI QR코드

DOI QR Code

Minimizing the Average Distance of Separated Points on the Plane in the L1-Distance

  • Kim, Jae-Hoon (Division of Computer Engineering, Pusan University of Foreign Studies)
  • Received : 2011.11.09
  • Accepted : 2011.12.20
  • Published : 2012.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Given separated points divided by a line, called a wall, in a plane, we aim to make a gate in the wall to connect the separated points to each other. In this setting, the problem is to find a location for the gate that minimizes the average distance between the points. The problem is a variant of the well-known facility location problem, which is extensively studied in the fields of operations research, location theory, theoretical computer science, and so on. In this paper, we consider the $L^1$-distance of the points in the plane. The points are projected onto the wall and so the problem is transformed to a proximity problem of points on a line. Then it is shown that the transformed problem is related to the weighted median problem of points on the line. Therefore, we obtain an O(n log n)-time algorithm to solve our problem.

Keywords

References

  1. F. R. K. Chung, "The average distance and the independence number," Journal of Graph Theory, vol. 12, no. 2, pp. 229-235, 1988. https://doi.org/10.1002/jgt.3190120213
  2. P. Dankelmann, "Computing the average distance of an interval graph," Information Processing Letters, vol. 48, no. 6, pp. 311- 314, 1993. https://doi.org/10.1016/0020-0190(93)90174-8
  3. S. L. Hakimi, "Location theory," in Handbook of Discrete and Combinatorial Mathematics. Boca Raton: CRC Press, pp. 986-995, 2000.
  4. P. Indyk, "Sublinear time algorithms for metric space problems," Proceedings of the 31st Annual ACM Symposium on the Theory of Computing, Atlanta: GA, pp. 428-434, 1999.
  5. P. Bose, A. Maheshwari, and P. Morin, "Fast approximations for sums of distances, clustering and Fermat-Weber problem," Computational Geometry: Theory and Applications, vol. 24, no. 3, pp. 135-146, 2003. https://doi.org/10.1016/S0925-7721(02)00102-5
  6. R. Hassin and A. Tamir, "Improved complexity bounds for location problems on the real line," Operations Research Letters, vol. 10, no. 7, pp. 395-402, 1991. https://doi.org/10.1016/0167-6377(91)90041-M
  7. N. Megiddo and K. J. Supowit, "On the complexity of some common geometric location problems," SIAM Journal on Computing, vol. 13, no. 1, pp. 182-196, 1984. https://doi.org/10.1137/0213014
  8. S. Arora, P. Raghavan, and S. Rao, "Approximation schemes for Euclidean k-medians and related problems," Proceedings of the 30th Annual ACM Symposium on Theory of Computing, Dallas: TX, pp. 106-113, 1998.
  9. N. Megiddo, "Linear-time algorithms for linear programming in $R^{3}$ and related problems," SIAM Journal on Computing, vol. 12, no. 4, pp. 759-776, 1983. https://doi.org/10.1137/0212052
  10. P. K. Agarwal, M. Sharir, and S. Toledo, "An efficient multi-dimensional searching technique and its applications," Duke University, Durham: NC, Technical Report CS-1993-20, 1993.
  11. B. Chazelle and J. Matousek, "On linear-time deterministic algorithms for optimization problems in fixed dimension," Journal of Algorithms, vol. 21, no. 3, pp. 579-597, 1996. https://doi.org/10.1006/jagm.1996.0060
  12. Z. Drezner, "The planar two-center and two-median problems," Transportation Science, vol. 18, no. 4, pp. 351-361, 1984. https://doi.org/10.1287/trsc.18.4.351
  13. D. Eppstein, "Dynamic three-dimensional linear programming," ORSA Journal on Computing, vol. 4, no. 4, pp. 360-368, 1992. https://doi.org/10.1287/ijoc.4.4.360
  14. J. Hershberger, "A faster algorithm for the two-center decision problem," Information Processing Letters, vol. 47, no. 1, pp. 23-29, 1993. https://doi.org/10.1016/0020-0190(93)90153-Z
  15. P. K. Agarwal and M. Sharir, "Planar geometric location problems," Algorithmica, vol. 11, pp. 185-195, 1994. https://doi.org/10.1007/BF01182774
  16. M. J. Katz and M. Sharir, "An expander-based approach to geometric optimization," Proceedings of the 9th Annual Symposium on Computational Geometry, San Diego: CA, pp. 198-207, 1993.
  17. M. Sharir, "A near-linear algorithm for the planar 2-center problem," Discrete and Computational Geometry, vol. 18, no. 2, pp. 125-134, 1997. https://doi.org/10.1007/PL00009311
  18. T. M. Chan, "More planar two-center algorithms," Computational Geometry, vol. 13, no. 3, pp. 189-198, 1999. https://doi.org/10.1016/S0925-7721(99)00019-X
  19. P. K. Agarwal and M. Sharir, "Efficient algorithms for geometric optimization", ACM Computing Surveys, vol. 30, no. 4, pp. 412-458, 1998. https://doi.org/10.1145/299917.299918
  20. G. N. Frederickson, "Parametric search and locating supply centers in trees," Proceedings of the 2nd Workshop on Algorithms and Data Structure, Ottawa, pp. 299-319, 1991.
  21. T. F. Gonzalez, "Clustering to minimize the maximum intercluster distance," Theoretical Computer Science, vol. 38, pp. 293-306, 1985. https://doi.org/10.1016/0304-3975(85)90224-5
  22. D. S. Hochbaum and D. B. Shmoys, "A unified approach to approximate algorithms for bottleneck problems," Journal of the ACM, vol. 33, no. 3, pp. 533-550, 1986. https://doi.org/10.1145/5925.5933
  23. T. Feder and D. Greene, "Optimal algorithms for approximate clustering," Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago: IL, pp. 434-444, 1988.
  24. P. K. Agarwal and C. M. Procopiuc, "Exact and approximation algorithms for clustering," Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco: CA, pp. 658-667, 1998.