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)


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.


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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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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