Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Abstract. Given a directed graph G = (V, E) and an integer k 1, a ktransitive-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 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 reconst...

• Type: Conference
• Year: 2010
• Where: APPROX
• Authors: Arnab Bhattacharyya, Elena Grigorescu, Madhav Jha, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff
• Keywords: Algorithms, APPROX 2010, Directed Graph, Efficient Local Monotonicity, Local Monotonicity Reconstructors