[论文解读] Tighter Estimates for epsilon-nets for Disks

Norbert Bus, Shashwat Garg+2 more|arXiv (Cornell University)|Jan 14, 2015

Computational Geometry and Mesh Generation参考文献 21被引用 5

一句话总结

本文提出了一种基于Delaunay三角剖分计算平面内圆盘$\epsilon$-网的构造性算法，将网大小的上界改进至$13.4/\epsilon$，显著优于此前最佳的$40/\epsilon$。实验结果表明，实际网大小约为$9/\epsilon$，在理论和实验层面均优于以往工作。

ABSTRACT

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set $P$ of points, and a set $\mathcal{D}$ of geometric objects in the plane, the goal is to compute a small-sized subset of $P$ that hits all objects in $\mathcal{D}$. In 1994, Bronniman and Goodrich made an important connection of this problem to the size of fundamental combinatorial structures called $ε$-nets, showing that small-sized $ε$-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives $O(1)$-factor approximation algorithms in $ ilde{O}(n)$ time for hitting sets for disks in the plane. This constant factor depends on the sizes of $ε$-nets for disks; unfortunately, the current state-of-the-art bounds are large -- at least $24/ε$ and most likely larger than $40/ε$. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than $40$. The best lower-bound is $2/ε$, which follows from the Pach-Woeginger construction for halfspaces in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem. The main goal of this paper is to improve the upper-bound to $13.4/ε$ for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of $ε$-nets for a variety of data-sets is lower, around $9/ε$.

研究动机与目标

• 为平面内圆盘的几何范围空间中$\epsilon$-网大小的最优上界与下界之间的巨大差距提供填补。
• 设计一种实用的、近乎线性时间的算法，用于计算小规模$\epsilon$-网，优于现有方案中使用的复杂计算工具。
• 提出一种基于Delaunay三角剖分的构造性方法，其结果在理论上优于以往方法。
• 利用多样化的现实世界和合成数据集，对理论界进行经验验证。
• 为圆盘的几何击中集问题提供更快、可证明性能良好的近似算法。

提出的方法

• 该算法基于Delaunay三角剖分，采用递归采样策略构造$\epsilon$-网，避免使用复杂的动态数据结构。
• 利用Delaunay三角剖分的几何特性，高效识别并采样能击中所有大圆盘的点。
• 通过涉及采样概率和几何打包常数的递归不等式，对网的期望大小进行分析。
• 使用参数$c_1$控制采样概率$\frac{c_1}{\epsilon n}$，当$c_1 = 12$时，理论网大小为$13.4/\epsilon$。
• 通过验证$\mathbb{E}[T(n)] \leq c n \log n$满足推导出的不等式，证明其期望运行时间的递推关系为$O(n \log n)$。
• 实现中使用CGAL进行Delaunay三角剖分和最近邻查询，采用单线程执行以保证公平比较。

实验结果

研究问题

• RQ1 平面内圆盘的$\epsilon$-网大小的上界能否低于$40/\epsilon$？
• RQ2 能否设计一种不依赖复杂动态数据结构的实用、近乎线性时间的$\epsilon$-网构造算法？
• RQ3 在现实世界和合成数据集中，$\epsilon$-网构造的实际性能如何？
• RQ4 理论界$13.4/\epsilon$是否能在实际中实现，且是否优于现有构造方法？
• RQ5 理论上的上界与目前已知的最佳下界$2/\epsilon$之间是否存在显著差距？

主要发现

• 本文为平面内圆盘的$\epsilon$-网大小建立了新的理论最优上界$13.4/\epsilon$，优于此前最佳的$40/\epsilon$。
• 该算法具有构造性，且期望时间复杂度为$O(n \log n)$，适用于近乎线性时间的击中集近似。
• 在多样数据集（MOPSI、KDDCUP、Europe、Birch3、Uniform、Gauss9）上的实验结果表明，平均$\epsilon$-网大小约为$9/\epsilon$，远低于理论界。
• 在采样概率$\frac{c_1}{\epsilon n}$中设置$c_1 = 12$，可得理论网大小$13.4/\epsilon$；而$c_1 = 7$时，实际网大小约为$9/\epsilon$。
• 基于Delaunay三角剖分的递归采样方法成功界定了期望网大小和运行时间，满足递推关系$\mathbb{E}[T(n)] \leq c n \log n$（其中$c \geq 2d$）。
• 实现代码为开源，并作为公共模块发布，展示了其在各类数据分布下的实际可行性与高效性。