最大空矩形和空盒的快速算法
Faster Algorithms for Largest Empty Rectangles and Boxes
摘要：
我们重新审视了计算几何中的一个经典问题：在$d$维度中的$n$给定点中找到最大体积的轴对齐空盒（在给定的边界框内）。以前，已知的最佳算法具有$d=2$的运行时间$O(n\log^2n)$（Aggarwal和Suri [SoCG'87]）和$d\ge 3$的接近$n^d$。我们描述更快的算法与运行时间（一）$d=2$的$O(n2^{O(\log^*n)}\log n)$，（二）$O(n^{2.5+o(1)})$时间$d=3$，和（三）$\widetilde{O}(n^{(5d+2)/6})$时间为任何常数$d\ge 4$。为了获得更高的维度的结果，我们适应和扩展以前的技术克利的措施问题，以优化某些目标函数的补充工会的orthant。
英文摘要：
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.
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