920 research outputs found
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
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