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.

翻译：已经证明，当用$n$进行归一化时，在大小为$\sigma$的字母表上，$d$个长度为$n$的随机字符串的最长公共子序列的期望长度会收敛于某个仅依赖于$d$和$\sigma$的常数。这些值被称为Chvátal-Sankoff常数，确定其精确值是一个众所周知的开放问题。对于某些$\sigma$和$d$的组合，已知其上界和下界，其中研究最广泛的$\sigma=2, d=2$情形的最佳下界和上界分别为$0.788071$和$0.826280$。基于先前用于下界估计常数的算法，我们通过实现运行时优化、并行化以及高效的内存读写方案，为$\sigma=2, d=2$情形获得了改进的下界$0.792665992$。此外，当字母表大小、字符串数量或两者均大于2时，我们几乎改进了所有先前报告的Chvátal-Sankoff常数的下界。