Searching 2D-Strings for Matching Frames

We study a natural type of repetitions in 2-dimensional strings. Such a repetition, called a matching frame, is a rectangular substring of size at least 2 × 2 with equal marginal rows and equal marginal columns. Matching frames first appeared in literature in the context of Wang tiles. We present two algorithms finding a matching frame with the maximum perimeter in a given n×m input string. The first algorithm solves the problem exactly in Õ(n^2.5) time (assuming n ≥ m). The second algorithm finds a (1 − ε)-approximate solution in Õ(^nm_ε4 ) time, which is near linear in the size of the input for constant ε. In particular, by setting ε = O(1) the second algorithm decides the existence of a matching frame in a given string in Õ(nm) time. Some technical elements and structural properties used in these algorithms can be of independent interest.