Transitive-Closure Spanners of the Hypercube and the Hypergrid [article]

Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff
2009 Electronic colloquium on computational complexity  
Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V, E H ) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners were introduced in [7] as a common abstraction for applications in access control, property testing and data structures. In this work we study the number of edges in the sparsest 2-TC-spanners for the directed hypercube {0, 1} d and hypergrid {1, 2, . . . ,
more » ... m} d with the usual partial order, , defined by: x 1 . . . x d y 1 . . . y d if and only if x i ≤ y i for all i ∈ {1, ..., d}. We show that the number of edges in the sparsest 2-TCspanner of the hypercube is 2 cd+Θ(log d) , where c ≈ 1.1620. We also present upper and lower bounds on the size of the sparsest 2-TC-spanner of the directed hypergrid. Our first pair of upper and lower bounds for the hypergrid is in terms of an expression with binomial coefficients. The bounds differ by a factor of O(d 2m ) and, in particular, give tight (up to a poly(d) factor) bounds for constant m. We also give a second set of bounds, which show that the number of edges in the sparsest 2-TC-spanner of the hypergrid is at most m d log d m and at least Ω max m d
dblp:journals/eccc/BhattacharyyaGJRW09 fatcat:aaaknf57qvh63eujjol2t2wbxm