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.
翻译:在计算几何中,我们重新研究一个典型的问题:在美元给定的美元值范围内,找到最大容量轴重的空框(在给定的捆绑框内),以美元为单位。以前,已知的最佳算法用美元=2美元(由Aggarwal和Suri[SoCG'87]支付),用美元接近美元=2美元。我们描述的是运行时间(一) 美元=2美元,(二) 美元=2.5+o(1)}美元,用美元运行时间为美元=3美元,和(三) 美元=4美元不变值(n ⁇ (5d+2)/6}),用美元运行时间为美元。为了获得更高维度的结果,我们调整和扩展了以前用于Klee计量问题的技术,以优化或数子组合的补充的某些客观功能。