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For an undirected tree with edges labeled by single letters, we consider its substrings, which are labels of the simple paths between two nodes. A palindrome is a word $w$ equal to its reverse $w^R$. We prove that the maximum number of distinct palindromic substrings in a tree of $n$ edges satisfies $\text{pal}(n)=O(n^{1.5})$. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who showed that $\text{pal}(n)=\Omega(n^{1.5})$. Hence, we settle the tight bound of $\Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for trees that are simple paths, the maximum palindromic complexity is exactly $n+1$. We also propose an $O(n^{1.5} \log^{0.5}{n})$-time algorithm reporting all distinct palindromes and an $O(n \log^2 n)$-time algorithm finding the longest palindrome in a tree.

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]., Comment: Accepted to ESA 2021

The notions of periodicity and repetitions in strings, and hence these of runs and squares, naturally extend to two-dimensional strings. We consider two types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a 2D-version of squares in standard strings). Amir et al. introduced 2D-runs, showed that there are $O(n^3)$ of them in an $n \times n$ 2D-string and presented a simple construction giving a lower bound of $\Omega(n^2)$ for their number (TCS 2020). We make a significant step towards closing the gap between these bounds by showing that the number of 2D-runs in an $n \times n$ 2D-string is $O(n^2 \log^2 n)$. In particular, our bound implies that the $O(n^2\log n + \textsf{output})$ run-time of the algorithm of Amir et al. for computing 2D-runs is also $O(n^2 \log^2 n)$. We expect this result to allow for exploiting 2D-runs algorithmically in the area of 2D pattern matching. A quartic is a 2D-string composed of $2 \times 2$ identical blocks (2D-strings) that was introduced by Apostolico and Brimkov (TCS 2000), where by quartics they meant only primitively rooted quartics, i.e. built of a primitive block. Here our notion of quartics is more general and analogous to that of squares in 1D-strings. Apostolico and Brimkov showed that there are $O(n^2 \log^2 n)$ occurrences of primitively rooted quartics in an $n \times n$ 2D-string and that this bound is attainable. Consequently the number of distinct primitively rooted quartics is $O(n^2 \log^2 n)$. Here, we prove that the number of distinct general quartics is also $O(n^2 \log^2 n)$. This extends the rich combinatorial study of the number of distinct squares in a 1D-string, that was initiated by Fraenkel and Simpson (J. Comb. Theory A 1998), to two dimensions. Finally, we show some algorithmic applications of 2D-runs. (Abstract shortened due to arXiv requirements.), Comment: To appear in the ESA 2020 proceedings

In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are given two strings $X$ and $Y$ of total length $n$, and we are asked to find a pair of maximal-length factors, one of $X$ and the other of $Y$, such that their Hamming distance is at most $k$. Thankachan et al. show that this problem can be solved in $\mathcal{O}(n \log^k n)$ time and $\mathcal{O}(n)$ space for constant $k$. We consider the LCF$_k$($\ell$) problem in which we assume that the sought factors have length at least $\ell$, and the LCF$_k$($\ell$) problem for $\ell=\Omega(\log^{2k+2} n)$, which we call the Long LCF$_k$ problem. We use difference covers to reduce the Long LCF$_k$ problem to a task involving $m=\mathcal{O}(n/\log^{k+1}n)$ synchronized factors. The latter can be solved in $\mathcal{O}(m \log^{k+1}m)$ time, which results in a linear-time algorithm for Long LCF$_k$. In general, our solution to LCF$_k$($\ell$) for arbitrary $\ell$ takes $\mathcal{O}(n + n \log^{k+1} n/\sqrt{\ell})$ time., Comment: submitted to CPM 2018

We investigate the function $L(h,p,q)$, called here the threshold function, related to periodicity of partial words (words with holes). The value $L(h,p,q)$ is defined as the minimum length threshold which guarantees that a natural extension of the periodicity lemma is valid for partial words with $h$ holes and (strong) periods $p,q$. We show how to evaluate the threshold function in $O(\log p + \log q)$ time, which is an improvement upon the best previously known $O(p+q)$-time algorithm. In a series of papers, the formulae for the threshold function, in terms of $p$ and $q$, were provided for each fixed $h \le 7$. We demystify the generic structure of such formulae, and for each value $h$ we express the threshold function in terms of a piecewise-linear function with $O(h)$ pieces., Comment: Full version of a paper accepted to LATA 2018