Sariel Har-Peled, Nirman Kumar
We study the Approximate Nearest Neighbor problem for metric spaces where the query points are constrained to lie on a subspace of low doubling dimension, while the data is high-dimensional. We show that this problem can be solved efficiently despite the high dimensionality of the data.
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Reverse nearest neighbor queries are defined as follows: Given an input point-set P, and a query point q, find all the points p in P whose nearest point in P U {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O(log n).
Incremental Planar Nearest Neighbor Queries with Optimal Query Time
本文提出一种支持最优 $O(\log n)$ 查询时间和 $O(\log^{1+\varepsilon}n)$ 插入时间的增量平面最近邻数据结构，通过利用问题几何性质的新分数级联技术实现，并可推广至其他半动态场景。
We start with a review of the pervasiveness of the nearest neighbor search problem and techniques used to solve it along with some experimental results. In the second chapter, we show reductions between two different classes of geo- metric proximity problems: the nearest neighbor problems to solve the Euclidean minimum spanning tree problem and the farthest neighbor problems to solve the k-centers problem. In the third chapter, we unify spatial partitioning trees un- der one framework the meta-tree. Finally, we propose a dual tree algorithm for Bichromatic Closest Pair and measure the complexi…
While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and $\ell_1$, few non-trivial algorithms are known for $\ell_p$ when ($2 < p < \infty$). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the first non-trivial approximation factor guarantees in this setting.
We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the \textit{zd-tree}. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size $n$ with bounded expansion constant and bounded ratio, the zd-tree can be built in $O(n)$ work with $O(n^ε)$ span for constant $ε<1$, and searching for the $k$-nearest neighbors …
We present PUFFINN, a parameterless LSH-based index for solving the $k$-nearest neighbor problem with probabilistic guarantees. By parameterless we mean that the user is only required to specify the amount of memory the index is supposed to use and the result quality that should be achieved. The index combines several heuristic ideas known in the literature. By small adaptions to the query algorithm, we make heuristics rigorous. We perform experiments on real-world and synthetic inputs to evaluate implementation choices and show that the implementation satisfies the quality guarantees while be…
This paper studies a discrepancy-sensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric.
This paper proposes a new algorithm for reducing Approximate Nearest Neighbor problem to Approximate Near Neighbor problem. The advantage of this algorithm is that it achieves O(log n) query time. As a reduction problem, the uery time complexity is the times of invoking the algorithm for Approximate Near Neighbor problem. All former algorithms for the same reduction need polylog(n) query time. A box split method proposed by Vaidya is used in our paper to achieve the O(log n) query time complexity.
Nearest-neighbor search, which returns the nearest neighbor of a query point in a set of points, is an important and widely studied problem in many fields, and it has wide range of applications. In many of them, such as sensor databases, location-based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest-neighbor queries in a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for - computing all points that are nearest neighbors of a query po…
This paper provides the first solution to the kinetic reverse $k$-nearest neighbor (\rknn) problem in $\mathbb{R}^d$, which is defined as follows: Given a set $P$ of $n$ moving points in arbitrary but fixed dimension $d$, an integer $k$, and a query point $q\notin P$ at any time $t$, report all the points $p\in P$ for which $q$ is one of the $k$-nearest neighbors of $p$.
The problem of nearest-neighbor (NN) condensation aims to reduce the size of a training set of a nearest-neighbor classifier while maintaining its classification accuracy. Although many condensation techniques have been proposed, few bounds have been proved on the amount of reduction achieved. In this paper, we present one of the first theoretical results for practical NN condensation algorithms. We propose two condensation algorithms, called RSS and VSS, along with provable upper-bounds on the size of their selected subsets. Additionally, we shed light on the selection size of two other state…
The nearest neighbor problem is defined as follows: Given a set $P$ of $n$ points in some metric space $(X,D)$, build a data structure that, given any point $q$, returns a point in $P$ that is closest to $q$ (its "nearest neighbor" in $P$). The data structure stores additional information about the set $P$, which is then used to find the nearest neighbor without computing all distances between $q$ and $P$. The problem has a wide range of applications in machine learning, computer vision, databases and other fields. To reduce the time needed to find nearest neighbors and the amount of memory us…
Improving Reverse k Nearest Neighbors Queries
本文针对反向k最近邻查询问题，提出基于圆锥曲线的高效验证方法，并在VoR-tree上实现复杂度为O(k^{1.5}·log k)的算法，实验证明其效率显著优于现有最先进方法。
(Nearest) Neighbors You Can Rely On: Formally Verified k-d Tree Construction and Search in Coq
本文使用Coq证明助手形式化验证了k-d树的构建与K近邻搜索算法，提出了一种基于列表排列概念的直观方法来指定这些算法的属性。
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