Lower Bounds for Optimal Alignments of Binary Sequences
arXiv:0807.0051 · doi:10.1016/j.dam.2009.06.028
Abstract
In parametric sequence alignment, optimal alignments of two sequences are computed as a function of the penalties for mismatches and spaces, producing many different optimal alignments. Here we give a 3/(2^{7/3}π^{2/3})n^{2/3} +O(n^{1/3} \log n) lower bound on the maximum number of distinct optimal alignment summaries of length-n binary sequences. This shows that the upper bound given by Gusfield et. al. is tight over all alphabets, thereby disproving the "square root of n conjecture". Thus the maximum number of distinct optimal alignment summaries (i.e. vertices of the alignment polytope) over all pairs of length-n sequences is Theta(n^{2/3}).
12 pages, 3 figures, submitted to Discrete Applied Mathematics