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 \(O(n\log ^2n)\) for \(d=2\) (Aggarwal and Suri 1987) and near \(n^d\) for \(d\ge 3\). We describe faster algorithms with the following running times (where \(\varepsilon >0\) is an arbitrarily small constant and \(\widetilde{O}\) hides polylogarithmic factors): \(n2^{O({\log }^*n)}\log n\) for \(d=2\), \(O(n^{2.5+\varepsilon })\) for \(d=3\), and \(\widetilde{O}(n^{(5d+2)/6})\) 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.