Faster Algorithms for Largest Empty Rectangles and Boxes

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(nlog²n) for d = 2 (by Aggarwal and Suri [SoCG'87]) and near n^d for d ≥ 3. We describe faster algorithms with running time - O(n2^{O(log^*n)}log n) for d = 2, - O(n^{2.5+o(1)}) time for d = 3, and - O(n^{(5d+2)/6}) time for any constant d ≥ 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.