Improved Lower Bounds on the Expected Length of Longest Common Subsequences

It is known that, when normalized by $n$ as $n \rightarrow \infty$, the expected length of a longest common subsequence of $d$ random strings of length $n$ over an alphabet of size $\sigma$ converges to a constant that depends only on $d$ and $\sigma$. These values are known as the Chvátal-Sankoff constants, and determining their exact values is a major open problem. Building on previous algorithms for lower-bounding the constants, we implement parallelization, introduce significant runtime optimizations, and design an efficient recursive memory reading and writing scheme. With this, we obtain a new best 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.