Length of the longest common subsequence between overlapping words

TL;DR

This paper investigates the expected length of the longest common subsequence (LCS) between two overlapping random sequences, revealing it is approximately the maximum of the overlap length and the typical LCS length of independent sequences, with tail bounds provided.

Contribution

It introduces a new analysis of LCS length for overlapping sequences, connecting it to independent sequence LCS expectations and providing probabilistic tail bounds.

Findings

Expected LCS length approximates max(overlap, independent LCS)

Derived tail bounds for LCS length in overlapping sequences

Established theoretical relationship between overlap and LCS length

Abstract

Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected length of an LCS is approximately $\max(\ell, \mathbb{E}[L_n])$, where $L_n$ is the length of an LCS between two independent random sequences. We also obtain tail bounds on this quantity.