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

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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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  1. 1.

    Benkert, M., Wolff, A., Widmann, F., Shirabe, T.: The minimum Manhattan network problem: approximations and exact solutions. Comput. Geom. 35, 188–208 (2006)

  2. 2.

    Boltyanski, V., Martini, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Springer, Berlin (1997)

  3. 3.

    Brazil, M., Thomas, D.A., Weng, J.F., Zachariasen, M.: Canonical forms and algorithms for Steiner trees in uniform orientation metrics. Algorithmica 44, 281–300 (2006)

  4. 4.

    Brazil, M., Zachariasen, M.: Steiner trees for fixed orientation metrics. J. Glob. Optim. 43, 141–169 (2009)

  5. 5.

    Chepoi, V., Nouioua, K., Vaxès, Y.: A rounding algorithm for approximating minimum Manhattan networks. Theor. Comput. Sci. 390, 56–69 (2008) and APPROX-RANDOM 2005, pp. 40–51

  6. 6.

    Chin, F.Y.L., Guo, Z., Sun, H.: Minimum Manhattan network is NP-complete. In: Symposium on Computational Geometry, pp. 393–402 (2009)

  7. 7.

    Demaine, E.D., Harmon, D., Iacono, J., Kane, D.M., Patrascu, M.: The geometry of binary search trees. In: SODA, pp. 496–505 (2009)

  8. 8.

    Das, A., Gansner, E.R., Kaufmann, M., Kobourov, S., Spoerhase, J., Wolff, A.: Approximating minimum manhattan networks in higher dimensions. In: European Symposium on Algorithms (2011)

  9. 9.

    Du, D.-Z., Gao, B., Graham, R.L., Liu, Z.-C., Wan, P.-J.: Minimum Steiner trees in normed planes. Discrete Comput. Geom. 9, 351–370 (1993)

  10. 10.

    Durier, R., Michelot, C.: Sets of efficient points in normed space. J. Math. Anal. Appl. 117, 506–528 (1986)

  11. 11.

    Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier/North-Holland, Amsterdam (2000)

  12. 12.

    Fink, E., Wood, D.: Restricted-Orientation Convexity. Springer, Berlin (2004)

  13. 13.

    Fuchs, B., Schulze, A.: A simple 3-approximation of minimum Manhattan networks. In: CTW, pp. 26–29 (2008) (full version: Technical report zaik2008-570)

  14. 14.

    Guo, Z., Sun, H., Zhu, H.: A fast 2-approximation algorithm for the minimum Manhattan network problem. In: Proc. 4th International Conference on Algorithmic Aspects in Information Management. Lecture Notes in Computer Science, vol. 5034, pp. 212–223 (2008)

  15. 15.

    Guo, Z., Sun, H., Zhu, H.: Greedy construction of 2-approximation minimum Manhattan network. In: 19th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 5369, pp. 4–15 (2008)

  16. 16.

    Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Approximating a minimum Manhattan network. Nord. J. Comput. 8, 219–232 (2001) and APPROX-RANDOM 1999, pp. 28–37

  17. 17.

    Harmon, D.: New bounds on optimal binary search trees, Ph.D. thesis, MIT, 2006

  18. 18.

    Kato, R., Imai, K., Asano, T.: An improved algorithm for the minimum Manhattan network problem. In: 13th International Symposium on Algorithms and Computation. Lecture Notes Computer Science, vol. 2518, pp. 344–356 (2002)

  19. 19.

    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)

  20. 20.

    Lam, F., Alexanderson, M., Pachter, L.: Picking alignments from (Steiner) trees. J. Comput. Biol. 10, 509–520 (2003)

  21. 21.

    Nouioua, K.: Enveloppes de pareto et réseaux de Manhattan: caractérisations et algorithmes. Thèse de Doctorat en Informatique, Université de la Méditerranée, 2005

  22. 22.

    Seibert, S., Unger, W.: A 1.5-approximation of the minimal Manhattan network. In: 16th International Symposium on Algorihtms and Computation. Lecture Notes Computer Science, vol. 3827, pp. 246–255 (2005)

  23. 23.

    Schulze, A.: Approximation algorithms for network design problems. Doctoral thesis, Universität zu Köln, 2008

  24. 24.

    Thisse, J.F., Ward, J.E., Wendell, R.E.: Some properties of location problems with block and round norms. Oper. Res. 32, 1309–1327 (1984)

  25. 25.

    Thompson, A.C.: Minkowski Geometry, Encyclopedia of Mathematics and Applications, vol. 63. Cambridge University Press, Cambridge (1996)

  26. 26.

    Widmayer, P., Wu, Y.F., Wang, C.K.: On some distance problems in fixed orientations. SIAM J. Comput. 16, 728–746 (1987) and Symposium on Computational Geometry, 1985, pp. 186–195

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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).

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