Repetition-free longest common subsequence of random sequences

TL;DR

This paper investigates the asymptotic behavior of the repetition-free longest common subsequence (LCS) of two random sequences over a k-ary alphabet, identifying three regimes based on sequence length and alphabet size, and providing tail bounds for deviations.

Contribution

The study characterizes the limiting behavior of the repetition-free LCS in different growth regimes of sequence length and alphabet size, extending understanding in probabilistic combinatorics.

Findings

Identifies three distinct growth regimes for R based on n and k.

Establishes limiting behavior of R in each regime.

Provides tail bounds for large deviations of R.

Abstract

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.