Longest common subsequences between words of very unequal length

TL;DR

This paper investigates the expected length of the longest common subsequence between two words of very different lengths over a k-symbol alphabet, revealing how this expectation scales with parameters and providing bounds for large alphabets.

Contribution

It establishes the asymptotic behavior of the LCS length constant b3_{k,b5} and shows it is approximately 1 minus a quadratic function of b5, with evidence for its limit as k grows large.

Findings

b3_{k,b5} is of order 1 - cb5^2 uniformly in k and b5

For large k, b3_{k,b5} approaches 1 - frac{1}{4}b5^2

A matching lower bound for b3_{k,b5} is proved

Abstract

We consider the expected length of the longest common subsequence between two random words of lengths $n$ and $(1-\varepsilon)kn$ over $k$-symbol alphabet. It is well-known that this quantity is asymptotic to $\gamma_{k,\varepsilon} n$ for some constant $\gamma_{k,\varepsilon}$. We show that $\gamma_{k,\varepsilon}$ is of the order $1-c\varepsilon^2$ uniformly in $k$ and $\varepsilon$. In addition, for large $k$, we give evidence that $\gamma_{k,\varepsilon}$ approaches $1-\tfrac{1}{4}\varepsilon^2$, and prove a matching lower bound.