Shattering Thresholds for Random Systems of Sets, Words, and Permutations
Abstract
This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.
Keywords
Cite
@article{arxiv.1301.6371,
title = {Shattering Thresholds for Random Systems of Sets, Words, and Permutations},
author = {Anant P. Godbole and Samantha Pinella and Yan Zhuang},
journal= {arXiv preprint arXiv:1301.6371},
year = {2013}
}
Comments
25 pages