Transitive-Closure Spanners of the Hypercube and the Hypergrid
Electronic Colloquium on Computational Complexity (ECCC)(2009)
摘要
Given a directed graph G = (V,E) and an integer k � 1, a k-transitive-closure-spanner (k-TC- spanner) of G is a directed graph H = (V,EH) 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 s tructures. 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,...,m}d with the usual partial order, �, defined by: x1 ...xdy1 ...yd if and only if xiyi for all i 2 {1,...,d}. We show that the number of edges in the sparsest 2-TC- spanner of the hypercube is 2cd+�(logd), 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(d2m) 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 md logd m and at least �
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关键词
max,directed graph,binomial coefficient,transitive closure,property testing,upper and lower bounds,access control,partial order
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