Bibliographic Details
| Title: |
LOWER BOUNDS FOR LOCAL MONOTONICITY RECONSTRUCTION FROM TRANSITIVE-CLOSURE SPANNERS. |
| Authors: |
Bhattacharyya, Arnab, Grigorescu, Elena, Jha, Madhav, Jungh, Kyomin, Raskhodnikova, Sofya, Woodruff, David P. |
| Source: |
SIAM Journal on Discrete Mathematics; 2012, Vol. 26 Issue 2, p618-646, 29p, 8 Diagrams, 2 Charts |
| Subject Terms: |
Mathematical functions, Hypercubes, Directed graphs, Integers, Data structures, Algorithms |
| 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 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 ƒ: {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. [ABSTRACT FROM AUTHOR] |
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| Database: |
Complementary Index |