Improved Lower Bounds on the Expected Length of Longest Common Subsequences

TL;DR

This paper improves the lower bounds on the expected length of the longest common subsequence for multiple random strings, advancing the understanding of Chvátal-Sankoff constants through optimized algorithms and extensive computations.

Contribution

The authors develop optimized algorithms and computational methods to significantly improve lower bounds on Chvátal-Sankoff constants for various string and alphabet sizes.

Findings

Achieved a new lower bound of 0.792665992 for two binary strings.

Improved lower bounds for multiple strings and larger alphabets.

Enhanced computational techniques for estimating LCS bounds.

Abstract

It has been proven that, when normalized by $n$, the expected length of a longest common subsequence of $d$ random strings of length $n$ over an alphabet of size $\sigma$ converges to some constant that depends only on $d$ and $\sigma$. These values are known as the Chv\'{a}tal-Sankoff constants, and determining their exact values is a well-known open problem. Upper and lower bounds are known for some combinations of $\sigma$ and $d$, with the best lower and upper bounds for the most studied case, $\sigma=2, d=2$, at $0.788071$ and $0.826280$, respectively. Building off previous algorithms for lower-bounding the constants, we implement runtime optimizations, parallelization, and an efficient memory reading and writing scheme to obtain an improved lower bound of $0.792665992$ for $\sigma=2, d=2$. We additionally improve upon almost all previously reported lower bounds for the Chv\'{a}tal-Sankoff constants when either the size of alphabet, the number of strings, or both are larger than 2.