Sparse Long Blocks and the Micro-Structure of the Longest Common Subsequences

TL;DR

This paper investigates how long constant blocks in random strings are aligned in the Longest Common Subsequence problem, revealing different behaviors depending on alphabet size, with simulations supporting theoretical findings.

Contribution

It demonstrates that the alignment behavior of long blocks varies significantly between two-letter and three-or-more-letter alphabets, highlighting fundamental differences in optimal alignments.

Findings

Two-letter alphabets align long blocks mainly with matching symbols.

Three or more-letter alphabets align long blocks mainly with gaps.

Simulations confirm the theoretical difference in gap proportions for different alphabet sizes.

Abstract

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.