k近邻查询的动态数据结构

Sarita de Berg, F. Staals
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引用次数: 2

摘要

我们的目标是开发动态数据结构,支持在$O(f(n) + k)$时间内对平面上的一组$n$点进行$k$ -最近邻($k$ -NN)查询,其中$f(n)$是$n$的某个多对数函数。关键组件是一个通用查询算法,它允许我们同时找到分布在$t$子结构上的$k$ -NN,从而将查询时间中的$O(tk)$项减少到$O(k)$。将这种技术与对数方法相结合,允许我们将任何静态$k$ -NN数据结构转换为支持高效插入和查询的数据结构。对于完全动态的情况,该技术允许我们恢复之前声明的欧几里得距离的确定性、最坏情况$O(\log^2n/\log\log n +k)$查询时间,同时保留多对数更新时间。我们调整了这种数据结构,以支持在简单多边形中的一组站点之间的完全动态\emph{测地线}$k$ -NN查询。为此,我们设计了一个基于浅切割的,只删除$k$ -NN数据结构。更一般地说,我们为任何类型的距离函数获得一个动态平面$k$ -NN数据结构,我们可以为其构建垂直浅切割。我们将所有的方法应用于平面上的欧氏距离、测地线距离和一般的、常数复杂度的代数距离函数。
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Dynamic Data Structures for k-Nearest Neighbor Queries
Our aim is to develop dynamic data structures that support $k$-nearest neighbors ($k$-NN) queries for a set of $n$ point sites in the plane in $O(f(n) + k)$ time, where $f(n)$ is some polylogarithmic function of $n$. The key component is a general query algorithm that allows us to find the $k$-NN spread over $t$ substructures simultaneously, thus reducing an $O(tk)$ term in the query time to $O(k)$. Combining this technique with the logarithmic method allows us to turn any static $k$-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, $O(\log^2n/\log\log n +k)$ query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic \emph{geodesic} $k$-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only $k$-NN data structure. More generally, we obtain a dynamic planar $k$-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.
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