Optimal Algorithms for Constrained 1-Center Problems

  • Luis Barba
  • Prosenjit Bose
  • Stefan Langerman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n + m)logω) time, where ω =  min {n, m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m + nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n + m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(n logm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.


minimum enclosing circle facility location problems 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Luis Barba
    • 1
    • 2
  • Prosenjit Bose
    • 1
  • Stefan Langerman
    • 2
  1. 1.Carleton UniversityOttawaCanada
  2. 2.Université Libre de BruxellesBrusselsBelgium

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