Algorithmica

, Volume 71, Issue 1, pp 53–65 | Cite as

Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

Article
  • 144 Downloads

Abstract

We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k,t)-VFTS or simply k-VFTS), if for any subset SX with |S|≤k, it holds that dHS(x,y)≤td(x,y), for any pair of x,yXS.

For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m≥2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m,n)) by adding O(km) edges, where α is a functional inverse of the Ackermann’s function.

Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k2n) edges and maximum degree O(k2).

Keywords

Sparse fault-tolerant spanners Doubling dimension Bounded hop-diameter Bounded maximum degree 

References

  1. 1.
    Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.H.M.: Euclidean spanners: short, thin, and lanky. In: STOC, pp. 489–498 (1995) Google Scholar
  2. 2.
    Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: SODA, pp. 291–300 (1993) Google Scholar
  3. 3.
    Chan, T.-H.H., Gupta, A.: Small hop-diameter sparse spanners for doubling metrics. Discrete Comput. Geom. 41(1), 28–44 (2009) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chan, T.-H.H., Gupta, A., Maggs, B.M., Zhou, S.: On hierarchical routing in doubling metrics. In: SODA, pp. 762–771 (2005) Google Scholar
  5. 5.
    Chan, T.-H.H., Li, M., Ning, L.: Incubators vs zombies: fault-tolerant, short, thin and lanky spanners for doubling metrics (2012). arXiv:1207.0892 [cs.Os]
  6. 6.
    Chazelle, B.: Computing on a free tree via complexity-preserving mappings. Algorithmica 2, 337–361 (1987) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chechik, S., Langberg, M., Peleg, D., Roditty, L.: Fault-tolerant spanners for general graphs. In: STOC, pp. 435–444 (2009) Google Scholar
  8. 8.
    Czumaj, A., Zhao, H.: Fault-tolerant geometric spanners. Discrete Comput. Geom. 32(2), 207–230 (2004) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Das, G., Narasimhan, G.: A fast algorithm for constructing sparse Euclidean spanners. In: Symposium on Computational Geometry, pp. 132–139 (1994) Google Scholar
  10. 10.
    Dinitz, M., Krauthgamer, R.: Fault-tolerant spanners: better and simpler. In: PODC, pp. 169–178 (2011) Google Scholar
  11. 11.
    Dinitz, Y., Elkin, M., Solomon, S.: Shallow-low-light trees, and tight lower bounds for Euclidean spanners. In: FOCS, pp. 519–528 (2008) Google Scholar
  12. 12.
    Elkin, M., Solomon, S.: Narrow-shallow-low-light trees with and without steiner points. SIAM J. Discrete Math. 25(1), 181–210 (2011) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Elkin, M., Solomon, S.: Optimal Euclidean spanners: really short, thin and lanky. In: STOC (2013, to appear) Google Scholar
  14. 14.
    Gao, J., Guibas, L.J., Nguyen, A.: Deformable spanners and applications. In: SoCG, pp. 190–199 (2004) Google Scholar
  15. 15.
    Gottlieb, L.-A., Roditty, L.: An optimal dynamic spanner for doubling metric spaces. In: ESA, pp. 478–489 (2008) Google Scholar
  16. 16.
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: FOCS, pp. 534–543 (2003) Google Scholar
  17. 17.
    Har-Peled, S., Mendel, M.: Fast construction of nets in low dimensional metrics, and their applications. In: Symposium on Computational Geometry, pp. 150–158 (2005) Google Scholar
  18. 18.
    Levcopoulos, C., Narasimhan, G., Smid, M.H.M.: Efficient algorithms for constructing fault-tolerant geometric spanners. In: STOC, pp. 186–195 (1998) Google Scholar
  19. 19.
    Lukovszki, T.: New results of fault tolerant geometric spanners. In: WADS, pp. 193–204 (1999) Google Scholar
  20. 20.
    Narasimhan, G., Smid, M.H.M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007) CrossRefMATHGoogle Scholar
  21. 21.
    Solomon, S.: Fault-tolerant spanners for doubling metrics: better and simpler (2012). arXiv:1207.7040 [cs.Os]
  22. 22.
    Solomon, S., Elkin, M.: Balancing degree, diameter and weight in Euclidean spanners. In: ESA (1), pp. 48–59 (2010) Google Scholar
  23. 23.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22(2), 215–225 (1975) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceThe University of Hong KongPokfulamHong Kong

Personalised recommendations