Better Indexing for Rectangular Pattern Matching

TL;DR

This paper presents a simple, efficient indexing structure for two-dimensional rectangular pattern matching that improves query time and construction complexity, challenging previous lower bounds and extending results beyond square patterns.

Contribution

It introduces a novel indexing structure of size O(n log n) that supports fast queries for rectangular patterns, surpassing prior assumptions about lower bounds for such problems.

Findings

Supports queries in O(m + k log^ε n) time for any ε > 0

Constructed in near-linear time O~(n)

Size of the structure is O(n log n)

Abstract

We revisit the complexity of building, given a two-dimensional string of size $n$, an indexing structure that allows locating all $k$ occurrences of a two-dimensional pattern of size $m$. While a structure of size $\mathcal{O}(n)$ with query time $\mathcal{O}(m+k)$ is known for this problem under the additional assumption that the pattern is a square [Giancarlo, SICOMP 1995], a popular belief was that for rectangular patterns one cannot achieve such (or even similar) bounds, due to a lower bound for a certain natural class of approaches [Giancarlo, WADS 1993]. We show that, in fact, it is possible to construct a very simple structure of size $\mathcal{O}(n\log n)$ that supports such queries for any rectangular pattern in $\mathcal{O}(m+k\log^{\varepsilon}n)$ time, for any $\varepsilon>0$. Further, our structure can be constructed in $\tilde{\mathcal{O}}(n)$ time.