Given a two-dimensional array of symbols and a picture language over a finite alphabet, we investigate how to find rectangular subarrays that belong to the picture language. Two-dimensional pattern matching problems can be formulated by interpreting subarrays as matches and picture languages as patterns. We formulate four particular problems – finding maximum, minimum, any or all match(es) – and describe algorithms solving the problems for basic classes of picture languages, which include local picture languages and picture languages accepted by various types of deterministic two-dimensional finite automata such as four-way, three-way and two-way automata, on-line tessellation automata, and returning finite automata. We also prove that the pattern matching problems cannot be solved for the class of local picture languages in time linear in the input area unless the well known problem of triangle finding in a graph is solvable in quadratic time. This shows a fundamental difference between the complexity of one-dimensional and two-dimensional pattern matching.

给定一个二维符号数组和一个有限字母上的图片语言，我们研究如何找到属于图片语言的矩形子数组。可以通过将子数组解释为匹配，将图片语言解释为模式来制定二维模式匹配问题。我们提出了四个特定的问题-查找最大，最小，任何或所有匹配项-并描述了解决图片语言基本类问题的算法，这些基本类包括本地图片语言和各种类型的确定性二维有限元接受的图片语言自动方法，例如四向，三向和两向自动机，在线细分自动机和返回有限自动机。我们还证明，除非在图中的三角形可在二次时间内解决，否则不能在输入区域中以时间线性的方式解决模式匹配问题，这对于本地图片语言的类别而言是无法解决的。这显示了一维和二维模式匹配的复杂性之间的根本区别。