On the 2-Central Path Problem

  • Yongding Zhu
  • Jinhui Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


In this paper we consider the following 2-Central Path Problem (2CPP): Given a set of m polygonal curves \(\mathcal{P} =\{P_1,P_2,\ldots,P_m\}\) in the plane, find two curves P u and P l , called 2-central paths, that best represent all curves in \(\mathcal{P}\). Despite its theoretical interest and wide range of practical applications, 2CPP has not been well studied. In this paper, we first establish criteria that P u and P l ought to meet in order for them to best represent \(\mathcal{P}\). In particular, we require that there exists parametrizations f u (t) and f l (t) (t ∈ [a,b]) of P u and P l respectively such that the maximum distance from {f u (t), f l (t)} to curves in \(\mathcal{P}\) is minimized. Then an efficient algorithm is presented to solve 2CPP under certain realistic assumptions. Our algorithm constructs P u and P l in O(nmlog4 n 2 α(n) α(n)) time, where n is the total complexity of \(\mathcal{P}\) (i.e., the total number of vertices and edges), m is the number of curves in \(\mathcal{P}\), and α(n) is the inverse Ackermann function.Our algorithm uses the parametric search technique and is faster than arrangement-related algorithms (i.e. Ω(n 2)) when m ≪ n as in most real applications.


Event Point Total Complexity Input Curve Parallel Step Input Trajectory 
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  1. 1.
    Agarwal, P.K., Sharir, M.: Applications of parametric searching in geometric optimization. Journal of Algorithms 17, 292–318 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Computing Surveys 30, 412–458 (1998)CrossRefGoogle Scholar
  3. 3.
    Agarwal, P.K., Sharir, M., Welzl, E.: The discrete 2-center problem. In: Proceedings of the 13th Annual ACM Symposium Computation Geometry, pp. 147–155 (1997)Google Scholar
  4. 4.
    Buchin, K., Buchin, M., van Kreveld, M., Löffler, M., Silveira, R.I., Wenk, C., Wiratma, L.: Median Trajectories. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 463–474. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Chan, T.M.: More planar two-center algorithms. Computational Geometry Theory Application 13, 189–198 (1997)CrossRefGoogle Scholar
  6. 6.
    Drezner, Z.: The planar two-center and two-median problem. Transportation Science 18, 351–361 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Eppstein, D.: Faster construction of planar two-centers. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 131–138 (1997)Google Scholar
  8. 8.
    Goodrich, M.T.: 42 parallel algorithms in geometryGoogle Scholar
  9. 9.
    Hershberger, J.: A faster algorithm for the two-center decision problem. Information Processing Letters 47, 23–29 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Jaromczyk, J.W., Kowaluk, M.: A geometric proof of the combinatorial bounds for the number of optimal solutions to the 2-center euclidean problem. In: Proceedings of the 7th Canadadian Conference Computational Geometry, pp. 19–24 (1995)Google Scholar
  11. 11.
    Megiddo, N.: Combinatorial optimization with rational objective functions. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 1–12 (1978)Google Scholar
  12. 12.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. Journal ACM 30, 852–865 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Sharir, M.: A near-linear algorithm for the planar 2-center problem. In: Discrete Computational Geometry, pp. 147–155 (1996)Google Scholar
  14. 14.
    Singh, V., Mukherjee, L., Peng, J., Xu, J.: Ensemble clustering using semidefinite programming. In: Proceedings of the 21st Advances in Neural Information Processing Systems (2007)Google Scholar
  15. 15.
    Xu, L., Stojkovic, B., Zhu, Y., Song, Q., Wu, X., Sonka, M., Xu, J.: Efficient algorithms for segmenting globally optimal and smooth multi-surfaces. In: Proceedings of the 22nd Biennial International Conference on Information Processing in Medical Imaging, pp. 208–220 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yongding Zhu
    • 1
  • Jinhui Xu
    • 1
  1. 1.Department of Computer Science and EngineeringState University of New York at BuffaloBuffaloUSA

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