Efficient Recognition of Abelian Palindromic Factors and Associated Results

Abstract : A string is called a palindrome if it reads the same from left to right. In this paper we define the new concept of an abelian palindrome which satisfies the property of being abelian equivalent to some palindrome of the same length. The identification of abelian palindromes presents a novel combinatorial problem, with potential applications in filtering strings for palindromic factors. We present an algorithm to efficiently identify abelian palindromes, and additionally generate an abelian palindromic array, indicating the longest abelian palindrome at each location. Specifically, for an alphabet of size $$|\varSigma | \le \log _2(n)$$|Σ|≤log2(n) and after $$\mathcal {O}(n)$$O(n) time preprocessing using $$\mathcal {O}(n + |\varSigma |)$$O(n+|Σ|) space, we may determine if any factor is abelian palindromic in $$\mathcal {O}(1)$$O(1) time. Additionally, we may determine the abelian palindromic array in $$\mathcal {O}(|\varSigma |n)$$O(|Σ|n) time. We further specify the algorithmic complexity when this condition on alphabet size $$|\varSigma |$$|Σ| is relaxed.