A repetition-free Longest Common Subsequence (LCS) of two sequences and is an LCS of and where each symbol may appear at most once. Let denote the length of a repetition-free LCS of two sequences of symbols each one chosen randomly, uniformly, and independently over a -ary alphabet. We study the asymptotic, in and , behavior of and establish that there are three distinct regimes, depending on the relative speed of growth of and . For each regime we establish the limiting behavior of . In fact, we do more, since we actually establish tail bounds for large deviations of 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. (2010). 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 and , behavior of parameter .