Faster Algorithms for Largest Empty Rectangles and Boxes

TL;DR

This paper introduces faster algorithms for finding the largest empty axis-aligned box within a bounding box among given points, significantly improving computational efficiency in two and higher dimensions.

Contribution

It presents new algorithms with improved running times for the largest empty box problem in 2D and higher dimensions, extending techniques from Klee's measure problem.

Findings

Faster algorithm for 2D with $O(n2^{O( ext{log}^*n)} ext{log} n)$ time

Improved 3D algorithm with $O(n^{2.5+o(1)})$ time

Generalized approach for higher dimensions with $ ilde{O}(n^{(5d+2)/6})$ time

Abstract

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst $n$ given points in $d$ dimensions. Previously, the best algorithms known have running time $O(n\log^2n)$ for $d=2$ (by Aggarwal and Suri [SoCG'87]) and near $n^d$ for $d\ge 3$. We describe faster algorithms with running time (i) $O(n2^{O(\log^*n)}\log n)$ for $d=2$, (ii) $O(n^{2.5+o(1)})$ time for $d=3$, and (iii) $\widetilde{O}(n^{(5d+2)/6})$ time for any constant $d\ge 4$. To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee's measure problem to optimize certain objective functions over the complement of a union of orthants.