空间有效的平面扫描算法
Space-Efficient Plane-Sweep Algorithms
摘要：
我们介绍空间有效的平面扫描算法的基本平面几何问题。假设输入位于$n$项的只读数组中，并且可用工作空间为$Θ(s)$位，其中$\lg n \leq s \leq n \cdot \lg n$。三种技术，可以用来作为一般的工具，在不同的空间效率的算法中引入和采用我们的算法。特别是，我们给出了一个几乎最优的算法，找到一组$n$点，运行在$O(n^2/s + n \cdot \lg s)$时间之间的最接近的对。我们还给出了一个简单的算法来枚举运行在$O((n^2/s^{2/3}) \cdot \lg s + k)$时间内的$n$线段的交点，其中$k$是交点的数量。计数版本可以在$O((n^2/s^{2/3}) \cdot \lg s)$~time中求解。当线段轴平行时，给出了计算交点的$O((n^2/s) \cdot \lg^{4/3} s + n^{4/3} \cdot \lg^{1/3} n)$时间算法和计算交点的$O((n^2/s) \cdot \lg s \cdot \lg \lg s + n \cdot \lg s + k)$时间算法，其中$k$是交点的个数。最后给出了一个在$O((n^2/s + n \cdot \lg s) \cdot \sqrt{(n/s) \cdot \lg n})$时间内计算平行轴矩形的Klee测度的算法。
英文摘要：
We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of $n$ items and that the available workspace is $Θ(s)$ bits, where $\lg n \leq s \leq n \cdot \lg n$. Three techniques that can be used as general tools in different space-efficient algorithms are introduced and employed within our algorithms. In particular, we give an almost-optimal algorithm for finding the closest pair among a set of $n$ points that runs in $O(n^2/s + n \cdot \lg s)$ time. We also give a simple algorithm to enumerate the intersections of $n$ line segments that runs in $O((n^2/s^{2/3}) \cdot \lg s + k)$ time, where $k$ is the number of intersections. The counting version can be solved in $O((n^2/s^{2/3}) \cdot \lg s)$~time. When the segments are axis-parallel, we give an $O((n^2/s) \cdot \lg^{4/3} s + n^{4/3} \cdot \lg^{1/3} n)$-time algorithm for counting the intersections, and an algorithm for enumerating the intersections that runs in $O((n^2/s) \cdot \lg s \cdot \lg \lg s + n \cdot \lg s + k)$ time, where $k$ is the number of intersections. We finally present an algorithm that runs in $O((n^2/s + n \cdot \lg s) \cdot \sqrt{(n/s) \cdot \lg n})$ time to calculate Klee's measure of axis-parallel rectangles.
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