Dynamic all scores matrices for LCS score

TL;DR

This paper introduces algorithms for efficiently maintaining all scores matrices based on LCS scores under dynamic string modifications, optimizing update times by exploiting sparsity parameters.

Contribution

It presents novel algorithms for dynamic all scores matrices supporting character insertions with improved time complexities based on sparsity parameters.

Findings

Supports incremental updates to all scores matrices in O(Delta) and O(L) time.

Extends to support both prepend and append operations simultaneously in O(L log log L) time.

Exploits sparsity parameters L and Delta for optimized performance.

Abstract

The problem of aligning two strings A,B in order to determine their similarity is fundamental in the field of pattern matching. An important concept in this domain is the "all scores matrix" that encodes the local alignment comparison of two strings. Namely, let K denote the all scores matrix containing the alignment score of every substring of B with A, and let J denote the all scores matrix containing the alignment score of every suffix of B with every prefix of A. In this paper we consider the problem of maintaining an all scores matrix where the scoring function is the LCS score, while supporting single character prepend and append operations to A and N. Our algorithms exploit the sparsity parameters L=LCS(A,B) and Delta = |B|-L. For the matrix K we propose an algorithm that supports incremental operations to both ends of A in O(Delta) time. Whilst for the matrix J we propose an algorithm that supports a single type of incremental operation, either a prepend operation to A or an append operation to B, in O(L) time. This structure can also be extended to support both operations simultaneously in O(L log log L) time.