Distribution of the length of the longest common subsequence of two multi-state biological sequences

The length of the longest common subsequence (LCS) among two biological sequences has been used as a measure of similarity, and the application of this statistic is of importance in genomic studies. Even for the simple case of two sequences of equal length and composed of binary elements with equal state probabilities, the exact distribution of the length of the LCS remains an open question. This problem is also known as an NP-hard problem in computer science. Apart from combinatorial analysis, using the finite Markov chain imbedding technique, we derive the exact distribution for the length of the LCS between two multi-state sequences of different lengths. Numerical results are provided to illustrate the theoretical results.