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Steiner Transitive-Closure Spanners of Low-Dimensional Posets

  • Piotr Berman
  • Arnab Bhattacharyya
  • Elena Grigorescu
  • Sofya Raskhodnikova
  • David P. Woodruff
  • Grigory Yaroslavtsev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k -transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (V H , E H ) such that (1) V ⊆ V H and (2) for all vertices v,u ∈ V, the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TC-spanners of directed acyclic graphs or, equivalently, partially ordered sets. We study the relationship between the dimension of a poset and the size, denoted S k , of its sparsest Steiner k-TC-spanner.

We present a nearly tight lower bound on S 2 for d-dimensional directed hypergrids. Our bound is derived from an explicit dual solution to a linear programming relaxation of the 2-TC-spanner problem. We also give an efficient construction of Steiner 2-TC-spanners, of size matching the lower bound, for all low-dimensional posets. Finally, we present a nearly tight lower bound on S k for d-dimensional posets.

Keywords

Directed Graph Linear Programming Relaxation Full Version Dual Program Dual Linear Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ailon, N., Chazelle, B., Comandur, S., Liu, D.: Property-preserving data reconstruction. Algorithmica 51(2), 160–182 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atallah, M.J., Blanton, M., Fazio, N., Frikken, K.B.: Dynamic and efficient key management for access hierarchies. ACM Trans. Inf. Syst. Secur. 12(3) (2009)Google Scholar
  3. 3.
    Awerbuch, B.: Communication-time trade-offs in network synchronization. In: PODC, pp. 272–276 (1985)Google Scholar
  4. 4.
    Bhattacharyya, A., Grigorescu, E., Jha, M., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Lower bounds for local monotonicity reconstruction from transitive-closure spanners. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 448–461. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Transitive-closure spanners. In: Mathieu, C. (ed.) SODA, pp. 932–941. SIAM, Philadelphia (2009)Google Scholar
  6. 6.
    Chandra, A.K., Fortune, S., Lipton, R.J.: Lower bounds for constant depth circuits for prefix problems. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 109–117. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  7. 7.
    Chandra, A.K., Fortune, S., Lipton, R.J.: Unbounded fan-in circuits and associative functions. J. Comput. Syst. Sci. 30(2), 222–234 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Santis, A., Ferrara, A.L., Masucci, B.: New constructions for provably-secure time-bound hierarchical key assignment schemes. Theor. Comput. Sci. 407(1-3), 213–230 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dushnik, B., Miller, E.: Concerning similarity transformations of linearly ordered sets. Bulletin Amer. Math. Soc. 46, 322–326 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jha, M., Raskhodnikova, S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. Electronic Colloquium on Computational Complexity (ECCC) TR11-057 (2011)Google Scholar
  12. 12.
    Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Raskhodnikova, S.: Transitive-closure spanners: A survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 167–196. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Saks, M.E., Seshadhri, C.: Local monotonicity reconstruction. SIAM J. Comput. 39(7), 2897–2926 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM Journal on Matrix Analysis and Applications 3(3), 351–358 (1982)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Piotr Berman
    • 1
  • Arnab Bhattacharyya
    • 2
  • Elena Grigorescu
    • 3
  • Sofya Raskhodnikova
    • 1
  • David P. Woodruff
    • 4
  • Grigory Yaroslavtsev
    • 1
  1. 1.Pennsylvania State UniversityUSA
  2. 2.Massachusetts Institute of TechnologyUSA
  3. 3.Georgia Institute of TechnologyUSA
  4. 4.IBM Almaden Research CenterUSA

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