# Farthest-Point Queries with Geometric and Combinatorial Constraints

• Ovidiu Daescu
• Ningfang Mi
• Chan-Su Shin
• Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

## Abstract

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2 n) time. Our data structure needs O(n log n) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other.

## Keywords

Convex Hull Voronoi Diagram Computational Geometry Convex Polygon Query Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Agarwal, P.K., Matoušek, J., Suri, S.: Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Computational Geometry: Theory and Applications 1(4), 189–201 (1992)
2. 2.
Aggarwal, A., Guibas, L.J., Saxe, J.B., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete & Computational Geometry 4(6), 591–604 (1989)
3. 3.
Aggarwal, A., Kravets, D.: A linear time algorithm for finding all farthest neighbors in a convex polygon. Information Processing letters 31(1), 17–20 (1989)
4. 4.
Bespamyatnikh, S.: Computing closest points for segments. Int. J. Comput. Geom. Appl. 13(5), 419–438 (2003)
5. 5.
Bespamyatnikh, S., Snoeyink, J.: Queries with segments in Voronoi diagrams. Comput. Geom. Theory Appl. 16(1), 23–33 (2000)
6. 6.
Chen, D.Z., Daescu, O.: Space-efficient algorithms for approximating polygonal curves in two dimensional space. International Journal of Computational Geometry and Applications 13(2), 95–112 (2003)
7. 7.
Chen, D.Z., Daescu, O., Hershberger, J., Kogge, P.M., Snoeyink, J.: Polygonal path approximation with angle constraints. In: Proc. 12th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA 2001), pp. 342–343 (2001)Google Scholar
8. 8.
Cheong, O., Shin, C.-S., Vigneron, A.: Computing farthest neighbors on a convex polytope. Theoretical Computer Science 296, 47–58 (2003)
9. 9.
Cole, R., Yap, C.-K.: Geometric retrieval problems. In: Proc. 24th Ann. IEEE Symposium on Foundations of Computer Science (FOCS 1983), pp. 112–121 (1983)Google Scholar
10. 10.
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)
11. 11.
Daescu, O., Mi, N.: Polygonal path approximation: A query based approach. Computational Geometry: Theory and Applications 30(1), 41–58 (2005)
12. 12.
Edelsbrunner, H., Guibas, L.J., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM Journal on Computing 15, 317–340 (1986)
13. 13.
Gudmundsson, J., Haverkort, H., Park, S.-M., Shin, C.-S., Wolff, A.: Facility location and the geometric minimum-diameter spanning tree. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 146–160. Springer, Heidelberg (2002)
14. 14.
Gudmundsson, J., Haverkort, H., Park, S.-M., Shin, C.-S., Wolff, A.: Facility location and the geometric minimum-diameter spanning tree. Computational Geometry: Theory and Applications 27(1), 87–106 (2004)
15. 15.
Knuth, D.E.: Sorting and Searching. In: The Art of Computer Programming, vol. 3, Addison-Wesley, Reading (1973)Google Scholar
16. 16.
Matoušek, J.: Efficient partition trees. Discrete and Computational Geometry 8, 315–334 (1992)
17. 17.
Mitchell, J.S.B., O’Rourke, J.: Computational geometry column 42. SIGACT News 32(3), 63–72 (2001)
18. 18.
Mitra, P., Chaudhuri, B.: Efficiently computing the closest point to a query line. Pattern Recognition Letters 19(11), 1027–1035 (1998)
19. 19.
Mukhopadhyay, A.: Using simplicial partitions to determine a closest point to a query line. In: Proc. Canadian Conf. Comp. Geom (CCCG 2002), pp. 10–12 (2002)Google Scholar
20. 20.
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1990)Google Scholar
21. 21.
Vaidya, P.M.: An O(n logn) algorithm for the all-nearest-neighbors problem. Discrete and Computational Geometry 4, 101–115 (1989)

## Authors and Affiliations

• Ovidiu Daescu
• 1
• Ningfang Mi
• 2
• Chan-Su Shin
• 3
• Alexander Wolff
• 4
1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
2. 2.Department of Computer ScienceCollege of William and MaryWilliamsburgUSA
3. 3.School of Electronics and Information EngineeringHankuk University of Foreign StudiesKorea
4. 4.Department of Computer ScienceKarlsruhe UniversityKarlsruheGermany