Faster Algorithms for Largest Empty Rectangles and Boxes

Abstract

We revisit a classical problem in computational geometry: finding the largest-volume axisaligned 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 n) for d = 2 (by Aggarwal and Suri [SoCG’87]) and near n for d ≥ 3. We describe faster algorithms with running time • O(n2 ∗ n) log n) for d = 2, • O(n) time for d = 3, and • Õ(n) 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.