Hardness of Detecting Abelian and Additive Square Factors in Strings

TL;DR

This paper establishes the computational hardness of detecting Abelian and additive square factors in strings, showing that efficient algorithms would contradict the 3SUM conjecture, and provides conditional lower bounds and some upper bounds.

Contribution

It proves 3SUM-hardness for several problems related to Abelian and additive squares in strings, establishing their conditional optimality and extending existing techniques.

Findings

Proves 3SUM-hardness for detecting Abelian square factors.

Establishes conditional lower bounds for additive square detection.

Provides a subquadratic upper bound for Abelian square detection over constant alphabets.

Abstract

We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality of the state-of-the-art algorithms for finding such factors. Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of an odd half-length, (b) computing centers of all Abelian square factors, (c) detecting an additive square factor in a length-$n$ string of integers of magnitude $n^{\mathcal{O}(1)}$, and (d) a problem of computing a double 3-term arithmetic progression (i.e., finding indices $i \ne j$ such that $(x_i+x_j)/2=x_{(i+j)/2}$) in a sequence of integers $x_1,\dots,x_n$ of magnitude $n^{\mathcal{O}(1)}$. Problem (d) is essentially a convolution version of the AVERAGE problem that was proposed in a manuscript of Erickson. We obtain a conditional lower bound for it with the aid of techniques recently developed by Dudek et al. [STOC 2020]. Problem (d) immediately reduces to problem (c) and is a step in reductions to problems (a) and (b). In conditional lower bounds for problems (a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it using several string gadgets that include arbitrarily long Abelian-square-free strings. Our reductions also imply conditional lower bounds for detecting Abelian squares in strings over a constant-sized alphabet. We also show a subquadratic upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].