Covering a Set of Points with a Minimum Number of Lines

  • Magdalene Grantson
  • Christos Levcopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)


We consider the minimum line covering problem: given a set S of n points in the plane, we want to find the smallest number l of straight lines needed to cover all n points in S. We show that this problem can be solved in O(n log l) time if lO(log1 − εn), and that this is optimal in the algebraic computation tree model (we show that the Ω(nlog l) lower bound holds for all values of l up to \(O(\sqrt n)\)). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if \(l \in O(\sqrt[4]{n})\). For the case when l ∈ Ω(log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l.


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  1. 1.
    Ben-Or, M.: Lower Bounds for Algebraic Computation Trees. In: Proc. 15th Ann. ACM Symp. on Theory of Comp., pp. 80–86. ACM Press, New York (1983)Google Scholar
  2. 2.
    Edelsbrunner, H., Guibas, L., Stolfi, J.: Optimal Point Location in a Monotone Subdivision. SIAM J. Comput. 15, 317–340 (1986); Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1986)Google Scholar
  3. 3.
    Grantson, M., Levcopoulos, C.: Covering a Set of Points with a Minimum Number of Lines. Technical Report LU-CS-TR:2005-236, ISSN 1650-1276 Report 156, Also at:
  4. 4.
    Guibas, L., Overmars, M., Robert, J.: The Exact Fitting Problem in Higher Dimensions. Computational Geometry: Theory and Applications 6, 215–230 (1996)MATHMathSciNetGoogle Scholar
  5. 5.
    Johnson, D.: Approximation Algorithms for Combinatorial Problems. J. of Comp. Syst. Sci. 9, 256–278 (1974)MATHCrossRefGoogle Scholar
  6. 6.
    Kumar, V., Arya, S., Ramesh, H.: Hardness of Set Cover With Intersection 1. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 624–635. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Langerman, S., Morin, P.: Covering Things with Things. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 662–673. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Megiddo, N., Tamir, A.: On the Complexity of Locating Linear Facilities in the Plane. Operation Research Letters 1, 194–197 (1982)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Sarnak, N., Tarjan, R.E.: Planar Point Location Using Persistent Search Tree. Comm. ACM 29, 669–679 (1986)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Magdalene Grantson
    • 1
  • Christos Levcopoulos
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

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