近似最近邻距离
Approximating Nearest Neighbor Distances
摘要：
一些研究人员提出了使用非欧几里德度量的点集在欧几里德空间聚类噪声数据。几乎总是需要一个距离函数来识别同一聚类中点的接近程度，即使欧几里得聚类直径很大。因此，优选的是将较小的成本分配给保持靠近输入点的路径。在本文中，我们考虑最自然的度量与此属性，我们称之为最近邻度量。给定一个点集P和一条路径$γ$，我们的度量将$γ$中的每个点与其到P的距离进行充电。沿$γ$的总电荷确定其最近邻长度，其被正式定义为沿曲线到输入点的距离的积分。我们描述了一个$(3+\varepsilon)$ -近似算法和一个$(1+\varepsilon)$ -近似算法来计算最近邻度量。两种近似算法都工作在近似线性时间。前者使用稀疏图上的最短路径，只使用输入点。后者使用周围空间的稀疏样本，以找到良好的近似测地线路径。
英文摘要：
Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $γ$, our metric charges each point of $γ$ with its distance to P. The total charge along $γ$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.
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