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 an $n\times n$ array. The main contribution of this paper is the presentation of the first algorithm for locating all maximal 2D repetitions. The algorithm is efficient and straightforward, with runtime $O(n^2\log n+\rho )$, where $n^2$ is the size of the input array and ρ is the number of maximal 2D repetitions in the output.

中文翻译：

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二维最大重复

最大重复或字符串运行具有广泛的应用，因此已进行了广泛的研究。在本文中，我们将此概念扩展为二维，精确定义了最大的2D重复。我们提供了一个最大边界上可能发生的最大2D重复次数的初始界限。$ñ\times ñ$数组。本文的主要贡献是介绍了用于定位所有最大2D重复的第一个算法。该算法高效，简单，运行时$Ø（{ñ}^2\mathrm{日志}ñ+\rho ）$，在哪里 ${ñ}^2$是输入数组的大小，而ρ是输出中最大2D重复次数。