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Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

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Abstract

Let \({\mathcal{B}}\) be a centrally symmetric convex polygon of ℝ2 and ‖pq‖ be the distance between two points p,q∈ℝ2 in the normed plane whose unit ball is \({\mathcal{B}}\). For a set T of n points (terminals) in ℝ2, a \({\mathcal{B}}\)-network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of \({\mathcal{B}}\) and for every pair of terminals t i and t j , the network N(T) contains a shortest \({\mathcal{B}}\)-path between them, i.e., a path of length ‖t i t j ‖. A minimum \({\mathcal{B}}\) -network on T is a \({\mathcal{B}}\)-network of minimum possible length. The problem of finding minimum \({\mathcal{B}}\)-networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball \({\mathcal{B}}\) is a square (and hence the distance ‖pq‖ is the l 1 or the l -distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. 393–402, 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum \({\mathcal{B}}\)-network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56–69, 2008, and APPROX-RANDOM, pp. 40–51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.

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Correspondence to V. Chepoi.

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Catusse, N., Chepoi, V., Nouioua, K. et al. Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm. Algorithmica 63, 551–567 (2012). https://doi.org/10.1007/s00453-011-9560-z

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Keywords

  • Normed plane
  • Distance
  • Geometric network design
  • Manhattan network
  • Approximation algorithms