Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Abstract

Given a directed acyclic graph (DAG) Gn = (Vn;E), a functionon Gn is given by f : Vn ! R. Such a function is monotone iff(x) f(y) for all (x; y) 2 E. A local monotonicity reconstructor forGn, introduced by Saks and Seshadhri (SICOMP 2010), is a randomizedalgorithm that, given access to an oracle for an almost monotone functionf : Vn ! R on Gn, can quickly evaluate a related function g : Vn ! Rwhich is guaranteed to be monotone. Furthermore, the reconstructor canbe 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 inaccess control, property testing and data structures.In this paper, we show a connection between 2-TC-spanners of Gn andlocal monotonicity reconstructors for Gn. We show that an ecient localmonotonicity reconstructor for Gn implies a sparse 2-TC-spanner of Gn,providing a new technique for proving lower bounds for local monotonicityreconstructors. We present tight upper and lower bounds on the sizeof the sparsest 2-TC-spanners of the directed hypercube and hypergrid,DAGs which are very-well studied in this area. These bounds imply lowerbounds for local monotonicity reconstructors for the hypergrid (hypercube)that nearly match the known upper bounds.