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 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 δ-palindromes (i.e. palindromes with δ 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 + δ)) and space O(n · g).

Recent Advances of Palindromic Factorization

“…We start with maximal palindromic factorization presented by [2] in section 2. Later, we explain palindromic factorization with gaps, maximal palindromic factorization with errors and maximal palindromic factorization with gaps and errors presented by [1] in sections 3, 4 and 5.…”

“…In this section we present an an efficient solution to the Palindromic Factorization with Gaps problem has been introduced by Adamczyk et al [1] .…”

“…To solve the Palindromic Factorization with Gaps problem, Adamczyk et al [1] algorithm firstly modify each of the triples in G j to reflect the length constraint (m). More precisely, due to the length constraint, in each G j some triples will disappear completely, and at most one triple will get trimmed (i.e.…”

Off-line and on-line algorithms for closed string factorization

“…Additionally it was shown that the upper bound on the number of palindromic factors of a string coincides with the lower bound on the number of closed factors (see [3] and references therein). Thus the study of closed strings shows potential applications in connection with applications of palindromes [4]. On the algorithmic side Badkobeh et al in [2] presented (among others) an algorithm for the factorisation of a given string of length n into a sequence of longest closed factors (LCFs) in time and space O(n) and another algorithm for computing the longest closed factor starting at every position in the string in O(n log n log log n ) time and O(n) space.…”

“…In molecular biology, for instance, palindromic sequences are extensively studied: they are often distributed around promoters, introns, and untranslated regions, playing important roles in gene regulation and other cell processes (see e.g. [4] [10]).…”