# "Sliced" Subwindow Search: a Sublinear-complexity Solution to the Maximum Rectangle Problem

## TL;DR

This paper introduces a novel sublinear-complexity method for the maximum rectangle problem in matrices, significantly improving speed and memory efficiency for large matrices, especially in natural image applications.

## Contribution

The authors propose a new approach that interpolates between matrix sections to achieve sublinear time and memory complexity, outperforming existing methods in large-scale scenarios.

## Key findings

- Achieves 11x speedup over state-of-the-art methods
- Maintains 99% accuracy in natural image applications
- Effective for real-time and large-scale matrix problems

## Abstract

Considering a 2D matrix of positive and negative numbers, how might one draw a rectangle within it whose contents sum higher than all other rectangles'? This fundamental problem, commonly known the maximum rectangle problem or subwindow search, spans many computational domains. Yet, the problem has not been solved without demanding computational resources at least linearly proportional to the size of the matrix. In this work, we present a new approach to the problem which achieves sublinear time and memory complexities by interpolating between a small amount of equidistant sections of the matrix. Applied to natural images, our solution outperforms the state-of-the-art by achieving an 11x increase in speed and memory efficiency at 99% comparative accuracy. In general, our solution outperforms existing solutions when matrices are sufficiently large and a marginal decrease in accuracy is acceptable, such as in many problems involving natural images. As such, it is well-suited for real-time application and in a variety of computationally hard instances of the maximum rectangle problem.