# On the Variance of the Length of the Longest Common Subsequences in Random Words With an Omitted Letter

## TL;DR

This paper analyzes the variance of the longest common subsequence length between two random words, where one contains an extra letter with a certain probability, showing the variance grows linearly with word length.

## Contribution

It establishes that the variance of the LCS length is linear in the size of the words in a setting with an omitted letter and probabilistic letter distributions.

## Key findings

- Variance of LCS length is linear in n.
- The presence of an extra letter affects the variance growth.
- Results extend understanding of LCS behavior in non-uniform random words.

## Abstract

We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{\alpha_1, \alpha_2, \cdots, \alpha_m\}$, while the letters of the other word are i.i.d. drawn from $\{\alpha_1, \alpha_2, \cdots, \alpha_m, \alpha_{m+1}\}$, with probability $p > 0$ to be $\alpha_{m+1}$, and $(1-p)/m > 0$ for all the other letters. The order of the variance of this length is shown to be linear in $n$.