Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

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Given a directed acyclic graph (DAG) Gn = (Vn;E), a function on Gn is given by f : Vn ! R. Such a function is monotone if f(x)  f(y) for all (x; y) 2 E. A local monotonicity reconstructor for Gn, introduced by Saks and Seshadhri (SICOMP 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f : Vn ! R on Gn, can quickly evaluate a related function g : Vn ! R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. 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 are a common abstraction for applications in access control, property testing and data structures. In this paper, we show a connection between 2-TC-spanners of Gn and local monotonicity reconstructors for Gn. We show that an ecient local monotonicity reconstructor for Gn implies a sparse 2-TC-spanner of Gn, providing a new technique for proving lower bounds for local monotonicity reconstructors. We present tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid, DAGs which are very-well studied in this area. These bounds imply lower bounds for local monotonicity reconstructors for the hypergrid (hypercube) that nearly match the known upper bounds.
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Property Testing; Property Reconstruction; Monotone Functions; Spanners; Hypercube; Hypergrid

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CS-Conference Papers(학술회의논문)
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