Advertisement

Theory of Computing Systems

, Volume 61, Issue 4, pp 1011–1036 | Cite as

New Pairwise Spanners

  • Telikepalli Kavitha
Article
  • 135 Downloads

Abstract

Let G=(V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch β if for every (u,v)∈V×V, d i s t H (u,v)≤d i s t G (u,v) + β. The problem of computing sparse spanners with small stretch β is well-studied. Here we consider the following variant: we are given \(\mathcal {P} \subseteq V \times V\) and we seek a sparse subgraph H where d i s t H (u,v)≤d i s t G (u,v) + β for each \((u,v) \in \mathcal {P}\). That is, distances for pairs outside \(\mathcal {P}\) need not be well-approximated in H. Such a subgraph is called a pairwise spanner with additive stretch β and our goal is to construct such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: \(\mathcal {P} = S \times V\) and \(\mathcal {P} = S \times T\), where SV and TV, and show sparser pairwise spanners for these cases.

Keywords

Undirected graph Shortest paths Approximate distances Additive stretch Spanner 

Notes

Acknowledgments

I am grateful to the reviewers for their very helpful comments and suggestions.

References

  1. 1.
    Abboud, A., Bodwin, G.: Lower bound amplification theorems for graph spanners. In: Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 841–856 (2016)Google Scholar
  2. 2.
    Abboud, A., Bodwin, G.: The 4/3 additive spanner exponent is tight. To appear in the Proceedings of the 48th Annual Symposium on the Theory of Computing (STOC), 351–361 (2016)Google Scholar
  3. 3.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28 (4), 1167–1181 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discret. Comput. Geom. 9, 81–100 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Near-linear time construction of sparse neighborhood covers. SIAM J. Comput. 28(1), 263–277 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Awerbuch, B., Peleg, D.: Routing with polynomial communication-space trade-off. SIAM J. Comput. 5(2), 151–162 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Baswana, S., Kavitha, T.: Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM J. Comput. 39(7), 2865–2896 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: Additive spanners and (α,β)-spanners. ACM Transactions on Algorithms, 7(1) (2010)Google Scholar
  9. 9.
    Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in o(n 2 n) time. ACM Trans. Algorithms 2(4), 557–577 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Baswana, S., Sen, S.: A simple linear time algorithm for computing a (2k−1)-spanner of o(n 1+1/k) size in weighted graphs. Random Struct. Algoritm. 30(4), 532–563 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bodwin, G., Vassilevska Williams, V.: Better distance preservers and additive spanners. In: Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 855–872 (2016)Google Scholar
  12. 12.
    Bollobás, B., Coppersmith, D., Elkin, M.: Sparse distance preservers and additive spanners. SIAM J. Discret. Math. 19(4), 1029–1055 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chechik, S.: New additive spanners. In: Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 498–512 (2013)Google Scholar
  14. 14.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cohen, E.: Fast algorithms for constructing t-spanners and paths of stretch t. In: Proceedings of the 34th IEEE Symp. on Foundations of Computer Science (FOCS), pp 648–658 (1993)Google Scholar
  16. 16.
    Coppersmith, D., Elkin, M.: Sparse source-wise and pair-wise distance preservers. In: Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 660–669 (2005)Google Scholar
  17. 17.
    Cowen, L.J.: Compact routing with minimum stretch. J. Algorithms 28, 170–183 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cowen, L.J., Wagner, C.G.: Compact roundtrip routing in directed networks. J. Algorithms 50(1), 79–95 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cygan, M., Grandoni, F., Kavitha, T.: On pairwise spanners. In: Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science (STACS), pp 209–220 (2013)Google Scholar
  20. 20.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Elkin, M.: Computing almost shortest paths. ACM Trans. Algorithms 1(2), 283–323 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Elkin, M., Peleg, D.: (1 + 𝜖,β)-spanner construction for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gavoille, C., Peleg, D., Perennes, S., Raz, R.: Distance labeling in graphs. In: Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 210–219 (2001)Google Scholar
  24. 24.
    Halperin, S., Zwick, U.: Unpublished result (1996)Google Scholar
  25. 25.
    Kavitha, T.: New pairwise spanners. In: Proceedings of the 32th International Symposium on Theoretical Aspects of Computer Science (STACS), pp 513–526 (2015)Google Scholar
  26. 26.
    Kavitha, T., Varma, N.M.: Small stretch pairwise spanners. SIAM J. Discret. Math. 29(4), 2239–2254 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Knudsen, M.B.T.: Additive spanners: a simple construction. In: Proc. 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp 277–281 (2014)Google Scholar
  28. 28.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discret. Math. 13, 383–390 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Parter, M.: Bypassing Erdȯs’ girth conjecture: Hybrid spanners and sourcewise spanners. In: Proceedings of the 41st International Colloquium on Automata, Languages and Programming (ICALP), pp 608–619 (2014)Google Scholar
  30. 30.
    Peleg, D.: Proximity-preserving labeling schemes. J. Graph Theory 33(3), 167–176 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Peleg, D., Schaffer, A.A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18, 740–747 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pettie, S.: Low distortion spanners. ACM Transactions on Algorithms, 6(1) (2009)Google Scholar
  34. 34.
    Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: Proceedings of the 32nd Int. Colloq. on Automata, Languages, and Programming (ICALP), pp 261–272 (2005)Google Scholar
  35. 35.
    Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Proceedings of the 12th Annual European Symposium on Algorithms (ESA), pp 580–591 (2004)Google Scholar
  36. 36.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp 802–809 (2006)Google Scholar
  38. 38.
    Woodruff, D.P.: Additive spanners in nearly quadratic time. In: Proceedings of the 37th Int. Colloq. on Automata, Languages, and Programming (ICALP), pp 463–474 (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.MumbaiIndia

Personalised recommendations