Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Abstract

Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, 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 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 $f : \{1,2,\dots,m\}^d \to \mathbb{R}$, can quickly evaluate a related function $g : \{1,2,\dots,m\}^d \to \mathbb{R}$ 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, ...