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Considering a 2D matrix of positive and negative numbers, how might one drawa rectangle within it whose contents sum higher than all other rectangles'?This fundamental problem, commonly known the maximum rectangle problem orsubwindow search, spans many computational domains. Yet, the problem has notbeen solved without demanding computational resources at least linearlyproportional to the size of the matrix. In this work, we present a new approachto the problem which achieves sublinear time and memory complexities byinterpolating between a small amount of equidistant sections of the matrix.Applied to natural images, our solution outperforms the state-of-the-art byachieving an 11x increase in speed and memory efficiency at 99% comparativeaccuracy. In general, our solution outperforms existing solutions when matricesare sufficiently large and a marginal decrease in accuracy is acceptable, suchas in many problems involving natural images. As such, it is well-suited forreal-time application and in a variety of computationally hard instances of themaximum rectangle problem.

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