Local Alignment (with Concave Gap Weights)

Problem Definition

This work of Miller and Myers [11] deals with the problem of pairwise sequence alignment in which the distance measure is based on the gap penalty model. They proposed an efficient algorithm to solve the problem when the gap penalty is a concave function of the gap length.

Let X and Y be two strings (sequences) of alphabet $\Sigma$. The pairwise alignment $\mathcal{A}$ of X and Y maps X, Y into strings X ^′, Y ^′ that may contain spaces (not in $\Sigma$) such that (1) $\vert X^{{\prime}}\vert =\vert Y ^{{\prime}}\vert =\ell$; (2) removing spaces from X ^′ and Y ^′ returns X and Y , respectively; and (3) for any $1 \leq i \leq \ell$, X ^′[i] and Y ^′[i] cannot be both spaces where X ^′[i] denotes the ith character in X ^′.

To evaluate the quality of an alignment, there are many different measures proposed (e.g., edit distance, scoring matrix [12]). In this work, they consider the gap penalty model.

A gap in an alignment $\mathcal{A}$ of X and Yis a maximal...