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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Steiner Transitive-Closure Spanners of Low-Dimensional Posets
verfasst von : Piotr Berman, Arnab Bhattacharyya, Elena Grigorescu, Sofya Raskhodnikova, David P. Woodruff, Grigory Yaroslavtsev
Erschienen in: Automata, Languages and Programming
Verlag: Springer Berlin Heidelberg
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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.