Searching 2D-Strings for Matching Frames

TL;DR

This paper introduces algorithms for finding maximum perimeter matching frames in 2D strings, including an exact algorithm with $ ilde{O}(n^{2.5})$ complexity and a near-linear approximation, along with new structural insights.

Contribution

The paper presents the first algorithms for maximum perimeter matching frames in 2D strings, including an exact and a near-linear approximation method, with novel structural properties.

Findings

Exact algorithm with $ ilde{O}(n^{2.5})$ complexity

Near-linear $(1- ext{epsilon})$-approximation algorithm

Introduction of new structural properties of 2D strings

Abstract

We introduce the natural notion of a matching frame in a $2$-dimensional string. A matching frame in a $2$-dimensional $n\times m$ string $M$, is a rectangle such that the strings written on the horizontal sides of the rectangle are identical, and so are the strings written on the vertical sides of the rectangle. Formally, a matching frame in $M$ is a tuple $(u,d,\ell,r)$ such that $M[u][\ell ..r] = M[d][\ell ..r]$ and $M[u..d][\ell] = M[u..d][r]$. In this paper, we present an algorithm for finding the maximum perimeter matching frame in a matrix $M$ in $\tilde{O}(n^{2.5})$ time (assuming $n \ge m)$. Additionally, for every constant $\epsilon> 0$ we present a near-linear $(1-\epsilon)$-approximation algorithm for the maximum perimeter of a matching frame. In the development of the aforementioned algorithms, we introduce inventive technical elements and uncover distinctive structural properties that we believe will captivate the curiosity of the community.