2012/04/04 by Saba Amsalu, S. Amsalu, Amsalu, S. +6
Computer Science · Mathematics · Physics and Astronomy · #05A16 #60C05 #60F05 #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #math-ph #math.CO #math.MP #math.PR #msc:05A16 #msc:60C05 #msc:60F05
paper · pdf · doi:10.48550/arxiv.1204.1005
To appear: Journal of Statistical Physics
openalex publication_date 2012/04/04 · arxiv created 2014/01/30 · arxiv updated 2014/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider two random strings having the same length and generated by an iid sequence taking its values uniformly in a fixed finite alphabet. Artificially place a long constant block into one of the strings, where a constant block is a contiguous substring consisting only of one type of symbol. The long block replaces a segment of equal size and its length is smaller than the length of the strings, but larger than its square-root. We show that for sufficiently long strings the optimal alignment corresponding to a Longest Common Subsequence (LCS) treats the inserted block very differently depending on the size of the alphabet. For two-letter alphabets, the long constant block gets mainly aligned with the same symbol from the other string, while for three or more letters the opposite is true and the block gets mainly aligned with gaps. We further provide simulation results on the proportion of gaps in blocks of various lengths. In our simulations, the blocks are "regular blocks" in an iid sequence, and are not artificially inserted. Nonetheless, we observe for these natural blocks a phenomenon similar to the one shown in case of artificially-inserted blocks: with two letters, the long blocks get aligned with a smaller proportion of gaps; for three or more letters, the opposite is true. It thus appears that the microscopic nature of two-letter optimal alignments and three-letter optimal alignments are entirely different from each other.