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 ε 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. © 2011 Springer-Verlag.
CITATION STYLE
Berman, P., Bhattacharyya, A., Grigorescu, E., Raskhodnikova, S., Woodruff, D. P., & Yaroslavtsev, G. (2011). Steiner transitive-closure spanners of low-dimensional posets. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6755 LNCS, pp. 760–772). https://doi.org/10.1007/978-3-642-22006-7_64
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