Algorithms for Anti-Powers in Strings

TL;DR

This paper introduces algorithms for efficiently identifying anti-powers in strings, which are strings composed of distinct blocks, and establishes bounds on their maximum number within a string.

Contribution

It presents an optimal algorithm for locating all anti-powers of a given order in a string and proves a tight bound on their maximum number.

Findings

Optimal algorithm for anti-power detection

Lower bound on the number of anti-powers in a string

Anti-powers can be as numerous as Θ(n^2/k) in a string

Abstract

A string $S[1,n]$ is a power (or tandem repeat) of order $k$ and period $n/k$ if it can decomposed into $k$ consecutive equal-length blocks of letters. Powers and periods are fundamental to string processing, and algorithms for their efficient computation have wide application and are heavily studied. Recently, Fici et al. (Proc. ICALP 2016) defined an {\em anti-power} of order $k$ to be a string composed of $k$ pairwise-distinct blocks of the same length ($n/k$, called {\em anti-period}). Anti-powers are a natural converse to powers, and are objects of combinatorial interest in their own right. In this paper we initiate the algorithmic study of anti-powers. Given a string $S$, we describe an optimal algorithm for locating all substrings of $S$ that are anti-powers of a specified order. The optimality of the algorithm follows form a combinatorial lemma that provides a lower bound on the number of distinct anti-powers of a given order: we prove that a string of length $n$ can contain $\Theta(n^2/k)$ distinct anti-powers of order $k$.