Counting and Verifying Abelian Border Arrays of Binary Words

TL;DR

This paper investigates counting and verifying abelian border arrays of binary words, providing a formula for their number and an efficient algorithm for verification under the word-RAM model.

Contribution

It establishes the exact count of valid abelian border arrays and introduces an efficient verification algorithm for binary words.

Findings

Number of valid abelian border arrays of length n is 2^{n-1}

Verification reduces to computing the abelian border array of a binary word

Verification algorithm runs in O(n^2 / log^2 n) time

Abstract

In this note, we consider the problem of counting and verifying abelian border arrays of binary words. We show that the number of valid abelian border arrays of length \(n\) is \(2^{n-1}\). We also show that verifying whether a given array is the abelian border array of some binary word reduces to computing the abelian border array of a specific binary word. Thus, assuming the word-RAM model, we present an \(O\left(\frac{n^2}{\log^2n}\right)\) time algorithm for the abelian border array verification problem.