Repetition-free longest common subsequence of random sequences

A repetition free Longest Common Subsequence (LCS) of two sequences x and y is an LCS of x and y where each symbol may appear at most once. Let R denote the length of a repetition free LCS of two sequences of n symbols each one chosen randomly, uniformly, and independently over a k-ary alphabet. We study the asymptotic, in n and k, behavior of R and establish that there are three distinct regimes, depending on the relative speed of growth of n and k. For each regime we establish the limiting behavior of R. In fact, we do more, since we actually establish tail bounds for large deviations of R from its limiting behavior. Our study is motivated by the so called exemplar model proposed by Sankoff (1999) and the related similarity measure introduced by Adi et al. (2007). A natural question that arises in this context, which as we show is related to long standing open problems in the area of probabilistic combinatorics, is to understand the asymptotic, in n and k, behavior of parameter R.Comment: 15 pages, 1 figur

From Monomials to Words to graphs

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal generated by \sigma(I) in the free monoid is finitely generated. Whether there exists an ordering such that is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.Comment: 27 pages, 2 postscript figures, uses gastex.st