Lower bounds for local monotonicity reconstruction from transitive-closure spanners

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closurespanner (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 used in access control, property testing and data structures. We show a connection between 2-TC-spanners and local monotonicity filters. A local monotonicity filter, introduced by Saks and Seshadhri [SIAM J. Comput., pp. 2897- 2926], is a randomized algorithm that, given access to an oracle for an almost monotone function f : {1, 2, . . . ,m} ^d → R, can quickly evaluate a related function g : {1, 2, . . .,m} ^d → R which is guaranteed to be monotone. Furthermore, the filter can be implemented in a distributed manner. We show that an efficient local monotonicity filter implies a sparse 2-TC-spanner of the directed hypergrid, providing a new technique for proving lower bounds for local monotonicity filters. Our connection is, in fact, more general: an efficient local monotonicity filter 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 nearly tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid. These bounds imply stronger lower bounds for local monotonicity filters that nearly match the upper bounds of Saks and Seshadhri.