Two-Dimensional Maximal Repetitions

Maximal repetitions or runs in strings have a wide array of applications and thus have been extensively studied. In this paper, we extend this notion to 2-dimensions, precisely defining a maximal 2D repetition. We provide initial bounds on the number of maximal 2D repetitions that can occur in a matrix. The main contribution of this paper is the presentation of the first algorithm for locating all maximal 2D repetitions in a matrix. The algorithm is efficient and straightforward, with runtime O(n^2 log n log log n+ rho log n), where n^2 is the size of the input, and rho is the number of 2D repetitions in the output

The Number of Repetitions in 2D-Strings

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 proceeding