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.)

中文翻译：

---

二维字符串中的重复次数

字符串中的周期性和重复的概念，以及这些游程和平方的概念，自然地扩展到二维字符串。我们考虑二维字符串中的两种类型的重复：二维运行和四次（四次是标准字符串中正方形的二维版本）。阿米尔等人。引入了二维运行，表明在 $n \times n$ 二维字符串中有 $O(n^3)$，并提出了一个简单的构造，给出了 $\Omega(n^2)$ 的下界他们的编号（TCS 2020）。我们通过证明 $n \times n$ 二维字符串中的二维运行次数为 $O(n^2 \log^2 n)$，朝着缩小这些界限之间的差距迈出了重要的一步。特别是，我们的界限意味着 Amir 等人的算法的 $O(n^2\log n + \textsf{output})$ 运行时间。用于计算 2D 运行也是 $O(n^2 \log^2 n)$。我们希望这个结果允许在 2D 模式匹配领域在算法上利用 2D 运行。四次是由 Apostolico 和 Brimkov (TCS 2000) 引入的由 $2 \times 2$ 相同块（二维字符串）组成的二维字符串，其中四次它们仅表示原始根四次，即由原始块构建。这里我们的四次方程的概念更一般，类似于一维弦中的平方。Apostolico 和 Brimkov 表明，在 $n \times n$ 二维字符串中有 $O(n^2 \log^2 n)$ 个原始根四次出现，并且这个界限是可以达到的。因此，不同的原始根四次方程的数量是 $O(n^2 \log^2 n)$。在这里，我们证明不同的一般四次方程的数量也是 $O(n^2 \log^2 n)$。这将由 Fraenkel 和 Simpson (J. Comb. Theory A 1998) 发起的对一维弦中不同正方形数量的丰富组合研究扩展到二维。最后，我们展示了 2D 运行的一些算法应用。（由于 arXiv 要求，摘要已缩短。）