Lower Bounds for Optimal Alignments of Binary Sequences

TL;DR

This paper establishes a tight lower bound on the maximum number of distinct optimal alignment summaries for binary sequences, disproving a previous conjecture and showing the complexity scales as Theta(n^{2/3}).

Contribution

It provides the first tight lower bound matching known upper bounds for the number of optimal alignment summaries, resolving a longstanding conjecture.

Findings

Maximum number of optimal alignment summaries is Theta(n^{2/3})

Disproves the 'square root of n' conjecture for binary sequences

Shows tight bounds on alignment polytope vertices

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}\pi^{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}).