Longest Common Subsequences in Sets of Permutations
Combinatorics
2009-04-13 v1
Abstract
The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008) showed that when k>=3, every set of k permutations on [n] has a common subsequence of length at least n^{1/3}. We show that, surprisingly, this lower bound is asymptotically optimal for all constant values of k. Specifically, we show that for any k>=3 and n>=k^2 there exists a set of k permutations on [n] in which the longest common subsequence has length at most 32(kn)^{1/3}. The proof of the upper bound is constructive, and uses elementary algebraic techniques.
Cite
@article{arxiv.0904.1615,
title = {Longest Common Subsequences in Sets of Permutations},
author = {Paul Beame and Eric Blais and Dang-Trinh Huynh-Ngoc},
journal= {arXiv preprint arXiv:0904.1615},
year = {2009}
}
Comments
9 pages