空间有效的二维正交色范围计数
Space Efficient Two-Dimensional Orthogonal Colored Range Counting
摘要：
在二维正交着色范围计数问题中，我们对平面上的$n$着色点的集合$P$进行预处理，使得给定一个正交查询矩形，可以有效地计算该矩形中包含的点的不同颜色的数量。对于这个问题，我们设计了三个新的解决方案，和每一个的界限可以表示在某种形式的时空权衡。通过为这些解决方案设置适当的参数值，我们可以获得新的特定结果（空间是单词，$ε$是$(0,1)$中的任意常数）：** $O(n\lg^3 n)$空间和$O(\sqrt{n}\lg^{5/2} n \lg \lg n)$查询时间;** $O(n\lg^2 n)$空间和$O(\sqrt{n}\lg^{4+ε} n)$查询时间;** $O(n\frac{\lg^2 n}{\lg \lg n})$空间和$O(\sqrt{n}\lg^{5+ε} n)$查询时间;** $O(n\lg n)$空间和$O(n^{1/2+ε})$查询时间。一个已知的条件下界，这个问题的基础上，布尔矩阵乘法给出了一些证据的困难，实现近线性空间的解决方案，查询时间优于$\sqrt{n}$的多对数因子，使用纯组合的方法。因此，所有这些结果中的时间和空间界限是有效的。以前，在具有类似查询时间的解决方案中，最节省空间的解决方案使用$O(n\lg^4 n)$空间来在$O(\sqrt{n}\lg^8 n)$时间（SIAM）内回答查询。J. Comp.~ 2008年）。因此，上面列出的新结果都实现了空间效率的提高，而除了最后一个结果之外，所有结果都实现了查询时间的加速。
英文摘要：
In the two-dimensional orthogonal colored range counting problem, we preprocess a set, $P$, of $n$ colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space are in words and $ε$ is an arbitrary constant in $(0,1)$): ** $O(n\lg^3 n)$ space and $O(\sqrt{n}\lg^{5/2} n \lg \lg n)$ query time; ** $O(n\lg^2 n)$ space and $O(\sqrt{n}\lg^{4+ε} n)$ query time; ** $O(n\frac{\lg^2 n}{\lg \lg n})$ space and $O(\sqrt{n}\lg^{5+ε} n)$ query time; ** $O(n\lg n)$ space and $O(n^{1/2+ε})$ query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than $\sqrt{n}$ by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses $O(n\lg^4 n)$ space to answer queries in $O(\sqrt{n}\lg^8 n)$ time (SIAM. J. Comp.~2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.
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