Faster Algorithms for Largest Empty Rectangles and Boxes

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 $O(n{\log }^{2}n)$ 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) $O(n{2}^{O({\log }^{\ast }n)}\log n)$ for $d=2$, (ii) $O(n^{2.5+o(1)})$ time for $d=3$, and (iii) $\tilde{O}(n^{(5d+2)\mathrm{/}6})$ 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.