Palindromic Decompositions with Gaps and Errors

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 on-line algorithm for obtaining a palindromic decomposition of a string of length [Formula: see text] with the minimal total gap length in time [Formula: see text] and space [Formula: see text], where [Formula: see text] is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal [Formula: see text]-palindromes (i.e. palindromes with [Formula: see text] errors under the edit or Hamming distance) and [Formula: see text] allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time [Formula: see text] and space [Formula: see text]. Finally, we provide an implementation of our algorithms.