Otfried Cheong, Antoine Vigneron, Juyoung Yon
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).
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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.
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 investigate the problem of finding reverse nearest neighbors efficiently. Although provably good solutions exist for this problem in low or fixed dimensions, to this date the methods proposed in high dimensions are mostly heuristic. We introduce a method that is both provably correct and efficient in all dimensions, based on a reduction of the problem to one instance of $\e$-nearest neighbor search plus a controlled number of instances of {\em exhaustive $r$-\pleb}, a variant of {\em Point Location among Equal Balls} where all the $r$-balls centered at the data points that contain the query…
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 $p$.
On Practical Nearest Sub-Trajectory Queries under the Fréchet Distance
本文针对Fréchet距离下的子轨迹最近邻查询问题，提出层次简化树与自适应聚类查询算法，并设计贪心回溯算法高效求解Fréchet决策问题，实验证明其能有效剪枝搜索空间并显著降低计算量。
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.
We consider the recently introduced monochromatic reverse top-k queries which ask for, given a new tuple q and a dataset D, all possible top-k queries on D union {q} for which q is in the result. Towards this problem, we focus on designing indexes in two dimensions for repeated (or batch) querying, a novel but practical consideration. We present the insight that by representing the dataset as an arrangement of lines, a critical k-polygon can be identified and used exclusively to respond to reverse top-k queries. We construct an index based on this observation which has guaranteed worst-case qu…
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。
We describe fully retroactive dynamic data structures for approximate range reporting and approximate nearest neighbor reporting. We show how to maintain, for any positive constant $d$, a set of $n$ points in $\R^d$ indexed by time such that we can perform insertions or deletions at any point in the timeline in $O(\log n)$ amortized time. We support, for any small constant $ε>0$, $(1+ε)$-approximate range reporting queries at any point in the timeline in $O(\log n + k)$ time, where $k$ is the output size. We also show how to answer $(1+ε)$-approximate nearest neighbor queries for any point in …
Let $k$ be a nonnegative integer. In the approximate $k$-flat nearest neighbor ($k$-ANN) problem, we are given a set $P \subset \mathbb{R}^d$ of $n$ points in $d$-dimensional space and a fixed approximation factor $c > 1$. Our goal is to preprocess $P$ so that we can efficiently answer approximate $k$-flat nearest neighbor queries: given a $k$-flat $F$, find a point in $P$ whose distance to $F$ is within a factor $c$ of the distance between $F$ and the closest point in $P$. The case $k = 0$ corresponds to the well-studied approximate nearest neighbor problem, for which a plethora of results ar…
Learning Nearest-Neighbor Maps from Adaptive Queries
本文研究从自适应查询学习最近邻映射的问题，推广相关研究得到查询复杂度紧界，证明欧氏域中查询次数对$d$呈指数依赖，并改进欧氏球域的查询复杂度。
We address the problem of replicating a Voronoi diagram $V(S)$ of a planar point set $S$ by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in $S$; 2. the distance to and label(s) of the nearest site(s) in $S$; 3. a unique label for every nearest site in $S$. We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of $V(S)$ with $O(n)$ queries and $O(n \log^2 n)$ processing time. We also prove that queries of Type 3 can never exactly clone $V(S)$, but we show that with…
Given a set $S$ of $n$ points in $d$-dimensional Euclidean metric space $X$ and a small positive real number $ε$, we present an algorithm to preprocess $S$ and answer queries that require finding a set $S' \subseteq S$ of $ε$-approximate nearest neighbors (ANNs) to a given query point $q \in X$. The following are the characteristics of points belonging to set $S'$: - $\forall s \in S'$, $\exists$ a point $p \in X$ such that $|pq| \le ε$ and the nearest neighbor of $p$ is $s$, and - $\exists$ a $s' \in S'$ such that $s'$ is a nearest neighbor of $q$. During the preprocessing phase, from the Vor…
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…
Emerging location-based systems and data analysis frameworks requires efficient management of spatial data for approximate and exact search. Exact similarity search can be done using space partitioning data structures, such as Kd-tree, R*-tree, and Ball-tree. In this paper, we focus on Ball-tree, an efficient search tree that is specific for spatial queries which use euclidean distance. Each node of a Ball-tree defines a ball, i.e. a hypersphere that contains a subset of the points to be searched. In this paper, we propose Ball*-tree, an improved Ball-tree that is more efficient for spatial qu…
Let $P$ be a set of $n$ colored points. We develop efficient data structures that store $P$ and can answer chromatic $k$-nearest neighbor ($k$-NN) queries. Such a query consists of a query point $q$ and a number $k$, and asks for the color that appears most frequently among the $k$ points in $P$ closest to $q$. Answering such queries efficiently is the key to obtain fast $k$-NN classifiers. Our main aim is to obtain query times that are independent of $k$ while using near-linear space. We show that this is possible using a combination of two data structures. The first data structure allow us t…
Graph-Based Nearest-Neighbor Search without the Spread
本文提出了一种无需依赖扩散的最近邻搜索方法，通过结合外部数据结构和图结构实现高效的近似最近邻查询。
We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n …
Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given $k$ query points $Q=q_1,\cdots,q_k$, and a compatibility graph $G$ with vertices in $Q$, and the goal is to return data points $p_1,\cdots,p_k$ that minimize (i) the weighted sum of the distances from $q_i$ to $p_i$ and (ii) the weighted sum, over all edg…
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set of point sites $S$ in a static simple polygon $P$. Our data structure allows us to insert a new site in $S$, delete a site from $S$, and ask for the site in $S$ closest to an arbitrary query point $q \in P$. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in $P$. Our data structure supports queries in $O(\sqrt{n}\log n\log^2 m)$ time, where $n$ is the number of sites currently in $S$, and $m$ is the number of vert…
We present a scalable approach for range and $k$ nearest neighbor queries under computationally expensive metrics, like the continuous Fréchet distance on trajectory data. Based on clustering for metric indexes, we obtain a dynamic tree structure whose size is linear in the number of trajectories, regardless of the trajectory's individual sizes or the spatial dimension, which allows one to exploit low `intrinsic dimensionality' of data sets for effective search space pruning. Since the distance computation is expensive, generic metric indexing methods are rendered impractical. We present strat…
We study the k nearest neighbors problem in the plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals and with respect to a general family of distance functions including the L_p-norms and additively weighted Euclidean distances. For point sites in the Euclidean metric, after four decades of effort, an optimal data structure has recently been developed with O( n ) space, O( log n + k ) query time, and O( n log n ) preprocessing time. We develop a static data structure for the general setting with nearly optimal O(…
Report on Nearest Dominating Point Queries
本文研究多维空间中最近支配点查询问题，以餐厅评分为例，为每个点寻找支配它且距离最近的点，并在特定假设下改进算法。
In this paper, we study top-$k$ aggregate (or group) nearest neighbor queries using the weighted SUM operator under the $L_1$ metric in the plane. Given a set $P$ of $n$ points, for any query consisting of a set $Q$ of $m$ weighted points and an integer $k$, $ 1 \le k \le n$, the top-$k$ aggregate nearest neighbor query asks for the $k$ points of $P$ whose aggregate distances to $Q$ are the smallest, where the aggregate distance of each point $p$ of $P$ to $Q$ is the sum of the weighted distances from $p$ to all points of $Q$. We build an $O(n\log n\log\log n)$-size data structure in $O(n\log …
This is an algorithm for finding neighbors when the objects can freely move and have no predefined position. The query consists in finding neighbors for a center location and a given radius. Space is discretized in cubic cells. This algorithm introduces a direct spherical indexing that gives the list of all cells making up the query sphere, for any radius and any center location. It can additionally take in account both cyclic and non-cyclic regions of interest. Finding only the K nearest neighbors naturally benefits from the spherical indexing by minimally running through the sphere from cent…
We present the first sublinear memory sketch that can be queried to find the nearest neighbors in a dataset. Our online sketching algorithm compresses an N element dataset to a sketch of size $O(N^b \log^3 N)$ in $O(N^{(b+1)} \log^3 N)$ time, where $b < 1$. This sketch can correctly report the nearest neighbors of any query that satisfies a stability condition parameterized by $b$. We achieve sublinear memory performance on stable queries by combining recent advances in locality sensitive hash (LSH)-based estimators, online kernel density estimation, and compressed sensing. Our theoretical res…
In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We introduce a different model where we assume that the computation of a single distance is an expensive operation and consequently, the goal is to minimize the number of such distance queries. This model is motivated by metric spaces that appear in the context of topological data analysis. We consider two standard operations on metric spaces, namely the construction o…
Approximate near-neighbors search (\textsc{ANNS}) is a long-studied problem in computational geometry. %that has received considerable attention by researchers in the community. In this paper, we revisit the problem and propose the first data structure for curves under the (continuous) Fréchet distance in $\Reals^d$. Given a set $¶$ of $n$ curves of size at most $m$ each in $\Reals^d$, and a real fixed $δ>0$, we aim to preprocess $¶$ into a data structure so that for any given query curve $Q$ of size $k$, we can efficiently report all curves in $¶$ whose Fréchet distances to $Q$ are at most $δ…
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…
The {\em bottleneck distance} is a natural measure of the distance between two finite point sets of equal cardinality, defined as the minimum over all bijections between the point sets of the maximum distance between any pair of points put in correspondence by the bijection. In this work, we consider the problem of building a data structure $\mathbb{D}$ that indexes a collection of $m$ planar point sets (of varying sizes) and supports nearest bottleneck distance queries: given a query point set $Q$ of size $n$, we would like to find the point set(s) $P \in \mathbb{D}$ of size $n$ that are clos…
Optimal Planar $k$ Nearest Neighbors Queries for Additively Weighted Euclidean Distances and Convex Metrics
該研究針對加權歐氏距離與凸度量，構建了具O(n)空間、O(log n + k)查詢時間的最優k近鄰數據結構，推廣了淺分區定理，為非點位等通用距離函數的最優k近鄰查詢提供了基礎成果。
Efficient Nearest-Neighbor Search for Dynamical Systems with Nonholonomic Constraints
本文研究了非holonomic约束动态系统中最近邻搜索的复杂性，发现传统k-d树在子里曼度量下查询复杂度为Θ(N^p log N)，提出针对性改进策略提升效率。
We study the aggregate/group nearest neighbor searching for the MAX operator in the plane. For a set $P$ of $n$ points and a query set $Q$ of $m$ points, the query asks for a point of $P$ whose maximum distance to the points in $Q$ is minimized. We present data structures for answering such queries for both $L_1$ and $L_2$ distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the $L_1$ version, we build a data structure of O(n) size in $O(n\log n)$ time, such that each query can be answered in $O(m+\log n)$ time. For the $L_2$ version, we …
Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances
本文提出一种基于凸化方法的近似最近邻搜索数据结构，突破了以往仅适用于欧几里得度量的局限，在对数查询时间与近最优空间下支持多种非欧几里得距离，包括凸缩放距离与 Bregman 散度。
Point and Line Nearest-Neighbor Searching in 3-Space
本文针对三维空间中点、线、线段和三角形的最近邻搜索问题，提出了多种数据结构，在查询时间和存储大小上显著优于先前结果，并建立了权衡界限。
Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let $ s(x,y) $ be the similarity between two points $ x $ and $ y $. We define a similarity between a set $ Q$ and a point $ x $ by aggregating the similarities $ s(p,x) $ for all $ p\in Q $. For example, we can take $ s(p,x) $ to be the angular similarity between $ p $ and $ x $ (i.e., $1-{\angle (x,p)}/…
Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider nearest-neighbor queries against a set of line segments in $\mathbb{R}^d$, for constant dimension $d$. Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^d$ and an error parameter $\varepsilon > 0$, the objective is to build a data structure such that for any query point $q$, it is possible to return a line segment whose Euclidean distance from $q$ is at most $…
We propose a new data-structure, the generalized randomized kd forest, or kgeraf, for approximate nearest neighbor searching in high dimensions. In particular, we introduce new randomization techniques to specify a set of independently constructed trees where search is performed simultaneously, hence increasing accuracy. We omit backtracking, and we optimize distance computations, thus accelerating queries. We release public domain software geraf and we compare it to existing implementations of state-of-the-art methods including BBD-trees, Locality Sensitive Hashing, randomized kd forests, and…
Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm ε)-approximate kth-nearest neighbor queries in O(log n + 1/ε^d) time, where k and εare provided at query time. If k and εare provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large.
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set $S$ of point sites in a static simple polygon $P$. Our data structure allows us to insert a new site in $S$, delete a site from $S$, and ask for the site in $S$ closest to an arbitrary query point $q \in P$. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in $P$. Our data structure achieves polylogarithmic update and query times, and uses $O(n\log^3n\log m + m)$ space, where $n$ is the number of sites in $S$ and $…
Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the $\ell$-threshold probability and the expectation of the closest-pair distance of a realization of $S$. We propose the first algorithm to compute the $\ell$-threshold probability in $O(t+n\log n+ \min\{…
Efficient Nearest Neighbor Search Using Dynamic Programming
针对 KD-Tree 与 R-Tree 在远点查询或二维流形数据上的性能下降，提出用动态规划预计算编码 Voronoi 邻近结构的 DAG，实现 1–10 倍加速并适用于多类几何应用。
Fast Nearest Neighbor Search for $\ell_p$ Metrics
本文提出了一种在$\ell_p$空间中实现快速查询时间的随机数据结构，适用于大规模数据集的在线查询需求，改进了现有最先进方法的性能。
This paper introduces a new proximity graph, called the $k$-Semi-Yao graph ($k$-SYG), on a set $P$ of points in $\mathbb{R}^d$, which is a supergraph of the $k$-nearest neighbor graph ($k$-NNG) of $P$. We provide a kinetic data structure (KDS) to maintain the $k$-SYG on moving points, where the trajectory of each point is a polynomial function whose degree is bounded by some constant. Our technique gives the first KDS for the theta graph (\ie, $1$-SYG) in $\mathbb{R}^d$. It generalizes and improves on previous work on maintaining the theta graph in $\mathbb{R}^2$. As an application, we use the…
We introduce a new variant of the nearest neighbor search problem, which allows for some coordinates of the dataset to be arbitrarily corrupted or unknown. Formally, given a dataset of $n$ points $P=\{ x_1,\ldots, x_n\}$ in high-dimensions, and a parameter $k$, the goal is to preprocess the dataset, such that given a query point $q$, one can compute quickly a point $x \in P$, such that the distance of the query to the point $x$ is minimized, when ignoring the "optimal" $k$ coordinates. Note, that the coordinates being ignored are a function of both the query point and the point returned. We pr…
In the polytope membership problem, a convex polytope $K$ in $\mathbb{R}^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given any query point $q \in \mathbb{R}^d$, it is possible to determine efficiently whether $q \in K$. We consider this problem in an approximate setting. Given an approximation parameter $\varepsilon$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\varepsilon$ times $K$'s diameter. We assume that the dimension $d$ is fixed, and $K$ is presented as the intersection of $n$ halfspaces. Previous so…
Given a training set $P \subset \mathbb{R}^d$, the nearest-neighbor classifier assigns any query point $q \in \mathbb{R}^d$ to the class of its closest point in $P$. To answer these classification queries, some training points are more relevant than others. We say a training point is relevant if its omission from the training set could induce the misclassification of some query point in $\mathbb{R}^d$. These relevant points are commonly known as border points, as they define the boundaries of the Voronoi diagram of $P$ that separate points of different classes. Being able to compute this set o…
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