Upper bounds on the average edit distance between two random strings

TL;DR

This paper improves the upper bounds on the average edit distance between two random strings for small alphabets by adapting and enhancing Lueker's technique, also providing better lower bounds for the longest common subsequence length.

Contribution

It introduces an improved method based on Lueker's technique to tighten bounds on average edit distance and LCS length for small alphabets.

Findings

Improved upper bounds on average edit distance for small alphabets.

Enhanced lower bounds on the longest common subsequence length.

New implementation of Lueker's technique for better bounds.

Abstract

We study the average edit distance between two random strings. More precisely, we adapt a technique introduced by Lueker in the context of the average longest common subsequence of two random strings to improve the known upper bound on the average edit distance. We improve all the known upper bounds for small alphabets. We also provide a new implementation of Lueker technique to improve the lower bound on the average length of the longest common subsequence of two random strings for all small alphabets of size other than $2$ and $4$.