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An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs

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Abstract

A spanning tree T of a graph G is called a tree t-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we present an algorithm which constructs for an n-vertex m-edge unweighted graph G:

  • a tree (2⌊log2 n⌋)-spanner in O(mlogn) time, if G is a chordal graph (i.e., every induced cycle of G has length 3);

  • a tree (2ρ⌊log2 n⌋)-spanner in O(mnlog2 n) time or a tree (12ρ⌊log2 n⌋)-spanner in O(mlogn) time, if G is a graph admitting a Robertson-Seymour’s tree-decomposition with bags of radius at most ρ in G; and

  • a tree (2⌈t/2⌉⌊log2 n⌋)-spanner in O(mnlog2 n) time or a tree (6t⌊log2 n⌋)-spanner in O(mlogn) time, if G is an arbitrary graph admitting a tree t-spanner.

For the latter result we use a new necessary condition for a graph to have a tree t-spanner: if a graph G has a tree t-spanner, then G admits a Robertson-Seymour’s tree-decomposition with bags of radius at most ⌈t/2⌉ and diameter at most t in G.

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Notes

  1. When G is an unweighted graph, t can be assumed to be an integer.

  2. We realize that it is perfectly possible that the authors of [27] did not try to optimize the constants in their analysis, and it may be the case that a more careful analysis of their algorithm may lead to an improved leading constant.

References

  1. Abraham, I., Bartal, Y., Neiman, O.: Nearly tight low stretch spanning trees. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), Philadelphia, PA, USA, 25–28 October 2008, pp. 781–790. IEEE Comput. Soc., Los Alamitos (2008)

    Chapter  Google Scholar 

  2. Alon, N., Karp, R.M., Peleg, D., West, D.B.: A graph-theoretic game and its application to the k-server problem. SIAM J. Comput. 24, 78–100 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ausiello, G., D’Arti, A., Moscarini, M.: Chordality properties on graphs and minimal conceptual connections in semantic data models. J. Comput. Syst. Sci. 33, 179–202 (1986)

    Article  MATH  Google Scholar 

  4. Bǎdoiu, M., Demaine, E.D., Hajiaghayi, M.T., Sidiropoulos, A., Zadimoghaddam, M.: Ordinal embedding: approximation algorithms and dimensionality reduction. In: Proceedings of the International Workshop on Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX-RANDOM 2008), Boston, MA, USA, 25–27 August 2008. Lecture Notes in Computer Science, vol. 5171, pp. 21–34. Springer, Berlin (2008)

    Chapter  Google Scholar 

  5. Bǎdoiu, M., Indyk, P., Sidiropoulos, A.: Approximation algorithms for embedding general metrics into trees. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), New Orleans, Louisiana, 7–9 January 2007, pp. 512–521. SIAM, Philadelphia (2007)

    Google Scholar 

  6. Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. J. ACM 30, 479–513 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Berge, C.: Hypergraphs. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  8. Brandstädt, A., Chepoi, V., Dragan, F.: Distance approximating trees for chordal and dually chordal graphs. J. Algorithms 30, 166–184 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brandstädt, A., Dragan, F.F., Le, H.-O., Le, V.B.: Tree spanners on chordal graphs: complexity and algorithms. Theor. Comput. Sci. 310, 329–354 (2004)

    Article  MATH  Google Scholar 

  10. Brandstädt, A., Dragan, F.F., Le, H.-O., Le, V.B., Uehara, R.: Tree spanners for bipartite graphs and probe interval graphs. Algorithmica 47, 27–51 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  12. Buneman, A.: A characterization of rigid circuit graphs. Discrete Math. 9, 205–212 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cai, L., Corneil, D.G.: Tree spanners. SIAM J. Discrete Math. 8, 359–387 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chepoi, V., Dragan, F.: A note on distance approximating trees in graphs. Eur. J. Comb. 21, 761–766 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chepoi, V.D., Dragan, F.F., Estellon, B., Habib, M., Vaxes, Y., Xiang, Y.: Additive spanners and distance and routing labeling schemes for δ-hyperbolic graphs. Algorithmica 62, 713–732 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chepoi, V., Dragan, F.F., Estellon, B., Habib, M., Vaxès, Y.: Diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. In: Proceedings of the 24th Annual ACM Symposium on Computational Geometry (SoCG 2008), College Park, Maryland, USA, 9–11 June 2008, pp. 59–68. ACM, New York (2008)

    Google Scholar 

  17. Chepoi, V.D., Dragan, F.F., Newman, I., Rabinovich, Y., Vaxes, Y.: Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs. Discrete Comput. Geom. 47, 187–214 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chepoi, V.D., Dragan, F.F., Yan, C.: Additive sparse spanners for graphs with bounded length of largest induced cycle. Theor. Comput. Sci. 347, 54–75 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Demmer, M.J., Herlihy, M.: The arrow distributed directory protocol. In: Proceedings of the 12th International Symposium on Distributed Computing (DISC 1998), Andros, Greece, 24–26 September 1998. Lecture Notes in Computer Science, vol. 1499, pp. 119–133. Springer, Berlin (1998)

    Google Scholar 

  20. Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2000)

    Google Scholar 

  21. Dourisboure, Y., Dragan, F.F., Gavoille, C., Yan, C.: Spanners for bounded tree-length graphs. Theor. Comput. Sci. 383, 34–44 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Dourisboure, Y., Gavoille, C.: Tree-decompositions with bags of small diameter. Discrete Math. 307, 2008–2009 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Dragan, F.F., Fomin, F., Golovach, P.: Spanners in sparse graphs. J. Comput. Syst. Sci. 77, 1108–1119 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dragan, F.F., Köhler, E.: An approximation algorithm for the tree t-spanner problem on unweighted graphs via generalized chordal graphs. In: Proceedings of the International Workshop on Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX-RANDOM 2011), Princeton, NJ, USA, 17–19 August 2011. Lecture Notes in Computer Science, vol. 6845, pp. 171–183. Springer, Berlin (2011)

    Chapter  Google Scholar 

  25. Elkin, M., Emek, Y., Spielman, D.A., Teng, S.-H.: Lower-stretch spanning trees. SIAM J. Comput. 38, 608–628 (2008)

    Article  MathSciNet  Google Scholar 

  26. Elkin, M., Peleg, D.: Approximating k-spanner problems for k≥2. Theor. Comput. Sci. 337, 249–277 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Emek, Y., Peleg, D.: Approximating minimum max-stretch spanning trees on unweighted graphs. SIAM J. Comput. 38, 1761–1781 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fekete, S.P., Kremer, J.: Tree spanners in planar graphs. Discrete Appl. Math. 108, 85–103 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Gavoille, C., Katz, M., Katz, N.A., Paul, C., Peleg, D.: Approximate distance labeling schemes. In: Proceedings of the 9th Annual European Symposium on Algorithms (ESA 2001), Aarhus, Denmark, 28–31 August 2001. Lecture Notes in Computer Science, vol. 2161, pp. 476–488. Springer, Berlin (2001)

    Google Scholar 

  30. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory, Ser. B 16, 47–56 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  31. Gilbert, J.R., Rose, D.J., Edenbrandt, A.: A separator theorem for chordal graphs. SIAM J. Algebr. Discrete Methods 5, 306–313 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. Goldman, A.J.: Optimal center location in simple networks. Transp. Sci. 5, 212–221 (1971)

    Article  Google Scholar 

  33. Herlihy, M., Kuhn, F., Tirthapura, S., Wattenhofer, R.: Dynamic analysis of the arrow distributed protocol. Theory Comput. Syst. 39, 875–901 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kratsch, D., Le, H.-O., Müller, H., Prisner, E., Wagner, D.: Additive tree spanners. SIAM J. Discrete Math. 17, 332–340 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Liebchen, C., Wünsch, G.: The zoo of tree spanner problems. Discrete Appl. Math. 156, 569–587 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Lokshtanov, D.: On the complexity of computing tree-length. Discrete Appl. Math. 158, 820–827 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. Peleg, D.: Low stretch spanning trees. In: Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science (MFCS 2002). Lecture Notes in Computer Science, vol. 2420, pp. 68–80. Springer, Berlin (2002)

    Chapter  Google Scholar 

  38. Peleg, D., Reshef, E.: Low complexity variants of the arrow distributed directory. J. Comput. Syst. Sci. 63, 474–485 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. Peleg, D., Tendler, D.: Low stretch spanning trees for planar graphs. Technical Report MCS01-14, Weizmann Science Press of Israel, Israel (2001)

  40. Makowsky, J.A., Rotics, U.: Optimal spanners in partial k-trees. Manuscript

  41. Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  42. Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of the Thirteenth Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 2001), Heraklion, Crete, Greece, 4–6 July 2001, pp. 1–10. ACM, New York (2001)

    Chapter  Google Scholar 

  43. Walter, J.R.: Representations of rigid cycle graphs. Ph.D. Thesis, Wayne State University, Detroit (1972)

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Acknowledgements

We would like to thank reviewers for many useful comments.

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Authors and Affiliations

  1. Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH, 44242, USA

    Feodor F. Dragan

  2. Mathematisches Institut, Brandenburgische Technische Universität Cottbus, 03013, Cottbus, Germany

    Ekkehard Köhler

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  1. Feodor F. Dragan
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  2. Ekkehard Köhler
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Correspondence to Feodor F. Dragan.

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Dragan, F.F., Köhler, E. An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs. Algorithmica 69, 884–905 (2014). https://doi.org/10.1007/s00453-013-9765-4

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