# Palindromic Decompositions with Gaps and Errors

## TL;DR

This paper introduces algorithms for decomposing sequences into palindromes allowing gaps and errors, with applications in computational biology and combinatorics, optimizing for minimal total gap length under various constraints.

## Contribution

It extends existing palindrome decomposition algorithms to include gaps, errors, and length bounds, providing efficient solutions for complex sequence analysis.

## Key findings

- Algorithm for minimal gap length decomposition with gaps in O(n log n * g) time.
- Algorithm for -palindrome decomposition with gaps and errors in O(n (g + )) time.
- Applicable to biological sequence analysis and combinatorics on words.

## Abstract

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an algorithm for obtaining a palindromic decomposition of a string of length n with the minimal total gap length in time O(n log n * g) and space O(n g), where g is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal \delta-palindromes (i.e. palindromes with \delta errors under the edit or Hamming distance) and g allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time O(n (g + \delta)) and space O(n g).