Lower bounds for local monotonicity reconstruction from transitive-closure spanners

Arnab Bhattacharyya, Elena Grigorescu, Madhav Jha, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)618-646
Number of pages29
JournalSIAM Journal on Discrete Mathematics
Volume26
Issue number2
DOIs
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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