The Steiner centre of a set of points: Stability, eccentricity, and applications to mobile facility location

International Journal of Computational Geometry and Applications

Volumn 16, Issue 4, 2006, Pages 345-371

  Durocher, Stephane ^a   Kirkpatrick, David ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Approximation; Centre; Eccentricity; Euclidean centre; Facility location; Mobile facility location; Smallest enclosing circle; Stability; Steiner point

Indexed keywords

EID: 33746879883     PISSN: 02181959     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218195906002075     Document Type: Article

References (33)

1. P. Agarwal, M. Sharir and S. Toledo, An efficient multi-dimensional searching technique and its applications, Technical Report CS-1993-20, Department of Computer Science, Duke University, 1993.
2. 2 for moving points, in Proc. Canadian Conf. Computational Geometry, Vol. 17, 2005, to appear.
3. P. K. Agarwal, J. Gao and L. J. Guibas, Kinetic medians and kd-trees, in Proc. Tenth European Symp. Algorithms, Lecture Notes in Computer Science, Vol. 2461, 2002, pp. 5-16.
4. P. K. Agarwal, L. J. Guibas, J. Hershberger and E. Veach, Maintaining the extent of a moving point set, Discr. Comput. Geom. 26 (2001) 353-374.
5. P. K. Agarwal and S. Har-Peled, Maintaining approximate extent measures of moving points, in Proc. Symp. Discrete Algorithms (ACM Press, 2001), pp. 148-157.
6. H. Alt, O. Aichholzer and G. Rote, Matching shapes with a reference point, Int. J. Comput. Geom. Appl. 7 (1997) 349-363.
7. J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications, Vol. 2 (Birkhäuser, Boston, 1990).
8. J. Basch, L. Guibas and J. Hershberger, Data structures for mobile data, J. Algorith. 31(1) (1999) 1-28.
9. S. Bereg, B. Bhattacharya, D. Kirkpatrick and M. Segal, Competitive algorithms for mobile centers, Mob. Network. Appl. (2005), to appear.
10. S. Bespamyatnikh, B. Bhattacharya, D. Kirkpatrick and M. Segal, Mobile facility location, in Proc. Int. ACM Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, Vol. 4, 2000, pp. 46-53.
11. R. S. Bhatia, S. Guha, S. Khuller and Y. Sussmann, Facility location with dynamic distance functions, J. Combin. Optim. 2 (1998) 199-217.
12. S. Buchin, Lectures on Differential Geometry (World Scientific, Singapore, 1980).
13. M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, New Jersey, 1976).
14. S. Durocher and D. Kirkpatrick, The Gaussian centre of a set of mobile points, in Proc. Canadian Conf. Computational Geometry, Vol. 15, 2003, pp. 123-127.
15. 3, in Proc. Canadian Conf. Computational Geometry, Vol. 16, 2004, pp. 140-144.
16. 2, in Proc. Canadian Conf. Computational Geometry, Vol. 17, 2005, pp. 46-50.
17. J. Gao, L. J. Guibas, J. Hershberger, L. Zhang and A. Zhu, Discrete mobile centers, Discr. Comput. Geom. 30(1) (2003) 45-65.
18. B. Grünbaum, Convex Polytopes, 2nd edition (Wiley and Sons, NY, 1967).
19. L. J. Guibas, Kinetic data structures: a state of the art report, in Proc. Workshop on the Algorithmic Foundations of Robotics (A. K. Peters, Ltd., 1998), pp. 191-209.
20. J. Hershberger, Smooth kinetic maintenance of clusters, Comput. Geom.: Th. Appl. 31 (2005) 3-30.
21. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, The Mathematical Association of America, Washington, 1995.
22. C. Kimberling, Clark Kimberling's encyclopedia of triangle centers, http://faculty.evansville.edu/ck6/encyclopedia/ETC.html.
23. 3 and related problems, SIAM J. Comput. 4 (1983) 759-776.
24. T. J. Richardson, Approximation of planar convex sets from hyperplane probes, Discr. Comput. Geom. 18 (1997) 151-177.
25. G. T. Sallee, A valuation property of Steiner points, Mathematika 13 (1966) 76-82.
26. G. C. Shephard, Approximation problems for convex polyhedra, Mathematika 11 (1964) 9-18.
27. G. C. Shephard, Steiner points of convex polytopes, Canadian J. Math. 18 (1966) 1294-1300.
28. G. C. Shephard, A uniqueness theorem for the Steiner point of a convex region, J. London Math. Soc. 43 (1968) 439-444.
29. J. Steiner, Gesammelte Werke, Vol. 1 (Reimer, Berlin, 1881).
30. C. Thas, A generalization of the Lemoine point, Forum Geometricorum 3 (2003) 161-167.
31. E. Welzl, Smallest enclosing disks (balls and ellipsoids), in New Results and New Trends in Computer Science, ed. H. Maurer, Lecture Notes in Computer Science, Vol. 555 (Springer-Verlag, 1991), pp. 359-370.
32. G. O. Wesolowsky, Dynamic facility location, Manag. Sci. 19 (1973) 1241-1248.
33. G. O. Wesolowsky and W. G. Truscott, The multiperiod location-allocation of facilities, Manag. Sci. 22 (1975) 57-65.