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]. |
Year | DOI | Venue |
---|---|---|
2021 | 10.4230/LIPIcs.ESA.2021.77 | ESA |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 5 |
Name | Order | Citations | PageRank |
---|---|---|---|
Jakub Radoszewski | 1 | 624 | 50.36 |
wojciech rytter | 2 | 130 | 17.13 |
Juliusz Straszynski | 3 | 2 | 5.15 |
Tomasz Waleń | 4 | 706 | 39.62 |
Wiktor Zuba | 5 | 1 | 5.13 |