TY - GEN

T1 - Steiner transitive-closure spanners of low-dimensional posets

AU - Berman, Piotr

AU - Bhattacharyya, Arnab

AU - Grigorescu, Elena

AU - Raskhodnikova, Sofya

AU - Woodruff, David P.

AU - Yaroslavtsev, Grigory

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79960012949&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960012949&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-22006-7_64

DO - 10.1007/978-3-642-22006-7_64

M3 - Conference contribution

AN - SCOPUS:79960012949

SN - 9783642220050

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 760

EP - 772

BT - Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings

T2 - 38th International Colloquium on Automata, Languages and Programming, ICALP 2011

Y2 - 4 July 2011 through 8 July 2011

ER -