Discrete & Computational Geometry

, Volume 53, Issue 2, pp 296–326 | Cite as

Average Stretch Factor: How Low Does It Go?

Article

Abstract

In a geometric graph \(G\), the stretch factor between two vertices \(u\) and \(w\) is the ratio between the Euclidean length of the shortest path from \(u\) to \(w\) in \(G\) and the Euclidean distance between \(u\) and \(w\). The average stretch factor of \(G\) is the average stretch factor taken over all pairs of vertices in \(G\). We show that, for any constant dimension \(d\) and any set \(V\) of \(n\) points in \(\mathbb {R}^d\), there exists a geometric graph with vertex set \(V\) that has \(O(n)\) edges and that has average stretch factor \(1+ o_n(1)\). More precisely, the average stretch factor of this graph is \(1+O\big ((\log n/n)^{1/(2d+1)}\big )\). We complement this upper bound with a lower bound: There exist \(n\)-point sets in \(\mathbb {R}^2\) for which any graph with \(O(n)\) edges has average stretch factor \(1+\Omega (1/\sqrt{n})\). Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exist point sets \(V\) such that any graph with worst-case stretch factor \(1+o_n(1)\) has a superlinear number of edges.

Keywords

Spanners Average stretch Average dilation 

Notes

Acknowledgments

The authors of this paper are partly funded by NSERC and CFI. The authors are indebted to Shay Solomon for providing helpful feedback on an earlier version of this paper. This research was funded by NSERC and the Ontario Ministry of Research and Innovation.

References

  1. 1.
    Abraham, I. , Bartal, Y., Chan, H.T.-H., Dhamdhere, K., Gupta, A., Kleinberg, J.M., Neiman, O., Slivkins, A.: Metric embeddings with relaxed guarantees. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 83–100. IEEE Computer Society, Washington, DC (2005)Google Scholar
  2. 2.
    Abraham, I., Bartal, Y., Neiman, O.: Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion. In: Bansal, N., Pruhs, K., Stein, C. (eds.) Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 502–511. SIAM, Philadelphia (2007)Google Scholar
  3. 3.
    Aldous, D.J., Kendall, W.S.: Short-length routes in low-cost networks via Poisson line patterns. Adv. Appl. Probab 40(1), 1–21 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 3rd edn. Wiley, Hoboken (2008)CrossRefMATHGoogle Scholar
  5. 5.
    Bender, M. A., Farach-Colton, M.: The LCA problem revisited. In: Proceedings of Latin American Theoretical Informatics (LATIN 2000), pp. 88–94 (2000)Google Scholar
  6. 6.
    Bentley, J.L.: Multidimensional divide-and-conquer. Commun. ACM 23(5), 214–228 (1978)MathSciNetGoogle Scholar
  7. 7.
    Bose, P., Dujmović, V., Morin, P., Smid, M.: Robust geometric spanners. In: Proceedings of the Twenty-Ninth ACM Symposium on Computational Geometry (SoCG 2013). ACM Press, New York (2013)Google Scholar
  8. 8.
    Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 291–300 (1993)Google Scholar
  9. 9.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42(1), 67–90 (1995)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Carmi, P., Smid, M.: An optimal algorithm for computing angle-constrained spanners. J. Comput. Geom. 3, 196–221 (2012)MathSciNetGoogle Scholar
  11. 11.
    Elkin, M., Solomon, S.: Steiner shallow-light trees are exponentially lighter than spanning ones. In: Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science, pp. 373–382 (2011)Google Scholar
  12. 12.
    Eppstein, D.: Spanning trees and spanners. Technical Report 96-16, Department of Information and Computer Science, University of California, Irvine. http://www.ics.uci.edu/eppstein/pubs/Epp-TR-96-16.pdf (1996)
  13. 13.
    Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry (Chap. 9), pp. 425–461. Elsevier, Amsterdam (1999)Google Scholar
  14. 14.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer-Verlage, New York (2001)CrossRefGoogle Scholar
  15. 15.
    Lueker, G.S.: A data structure for orthogonal range queries. In: Proceedings of the 19th Annual Symposium on Foundations of Computer Science (FOCS’78), pp. 28–34. IEEE Computer Society, Long Beach, CA (1978)Google Scholar
  16. 16.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, New York (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Ruppert, J., Seidel, R.: Approximating the \(d\)-dimensional complete Euclidean graph. In: Proceedings of the 3rd Canadian Conference on Computational Geometry (CCCG 1991), pp. 207–210 (1991)Google Scholar
  18. 18.
    Salowe, J.S.: Constructing multidimensional spanner graphs. Int. J. Comput. Geom. Appl. 1, 99–107 (1991)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Solomon, S.: Personal Communication with M. Smid (2012)Google Scholar
  20. 20.
    Vaidya, P.M.: A sparse graph almost as good as the complete graph on points in \(K\) dimensions. Discrete Comput. Geom. 6, 369–381 (1991)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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