Abstract
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, E H ) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing and data structures.
We show a connection between 2-TC-spanners and local monotonicity reconstructors. A local monotonicity reconstructor, introduced by Saks and Seshadhri (SIAM Journal on Computing, 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f : [m]d →ℝ, can quickly evaluate a related function g : [m]d →ℝ which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. We show that an efficient local monotonicity reconstructor implies a sparse 2-TC-spanner of the directed hypergrid (hypercube), providing a new technique for proving lower bounds for local monotonicity reconstructors. Our connection is, in fact, more general: an efficient local monotonicity reconstructor for functions on any partially ordered set (poset) implies a sparse 2-TC-spanner of the directed acyclic graph corresponding to the poset.
We present tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid. These bounds imply tighter lower bounds for local monotonicity reconstructors that nearly match the known upper bounds.
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Bhattacharyya, A., Grigorescu, E., Jha, M., Jung, K., Raskhodnikova, S., Woodruff, D.P. (2010). Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_34
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DOI: https://doi.org/10.1007/978-3-642-15369-3_34
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