Lower bounds for local monotonicity reconstruction from transitive-closure spanners

  • Authors:

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

  • Affiliations:

  • Massachusetts Institute of Technology;Massachusetts Institute of Technology;Pennsylvania State University;Korea Advanced Institute of Science and Technology, Korea;Pennsylvania State University;IBM Almaden Research Center

  • Venue:
  • APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
  • Year:
  • 2010

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Abstract

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 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 reconstructors. Our connection is, in fact, more general: an efficient local monotonicity reconstructor 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 tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid. These bounds imply tighter lower bounds for local monotonicity reconstructors that nearly match the known upper bounds.