Improved Lower Bounds on the Expected Length of Longest Common Subsequences

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 Chvátal-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 Chvátal-Sankoff constants when either the size of alphabet, the number of strings, or both are larger than 2.