In this paper, we present algorithms and data structures for the top-k nearest neighbor searching where the input points are exact and the query point is uncertain under the L1 distance metric in the plane. The uncertain query point is represented by a discrete probability density function, and the goal is to return the top-k expected nearest neighbors, which have the smallest expected distances to the query point. Given a set of n exact points in the plane, we build an O(n log n log log n)-size data structure in O(n log n log log n) time, such that for any uncertain query point with m possible locations and any integer k with 1 k n, the top-k expected nearest neighbors can be found in O(mlogm + (k+m)log^2 n) time. Even for the special case where k = 1, our result is better than the previously best method (in PODS 2012), which requires O(n log^2 n) preprocessing time, O(n log^2 n) space, and O(m^2 log^3 n) query time. In addition, for the one-dimensional version of this problem, our approach can build an O(n)-size data structure in O(n log n) time that can support O(min mk,mlog m + k + log n) time queries and the query time can be reduced to O(k+m+log n) time if the locations of Q are given sorted. In fact, the problem is equivalent to the aggregate or group nearest neighbor searching with the weighted SUM as the aggregate distance function operator.
In the formulation of (SNN) @cite_14 , one considers the superseding relationship of each pair of input points: one supersedes the other if and only if it has probability more than 0.5 of being the nearest neighbor of the query point, where the probability computation is restricted to this pair of points. One can return the point, if such one exists, which supersedes all the others. Otherwise, one returns the minimal set @math of data points such that any data point in @math supersedes any data point not in @math .
@cite_14
2103895255 1608524621 2077178480 2013333366
This paper proposes a new problem, called superseding nearest neighbor search, on uncertain spatial databases, where each object is described by a multidimensional probability density function. Given a query point q, an object is a nearest neighbor (NN) candidate if it has a nonzero probability to be the NN of q. Given two NN-candidates o1 and o2, o1 supersedes o2 if o1 is more likely to be closer to q. An object is a superseding nearest neighbor (SNN) of q, if it supersedes all the other NN-candidates. Sometimes no object is able to supersede every other NN-candidate. In this case, we return the SNN-core-the minimum set of NN-candidates each of which supersedes all the NN-candidates outside the SNN-core. Intuitively, the SNN-core contains the best objects, because any object outside the SNN-core is worse than all the objects in the SNN-core. We show that the SNN-core can be efficiently computed by utilizing a conventional multidimensional index, as confirmed by extensive experiments. Given a point q, a reverse k nearest neighbor (RkNN) query retrieves all the data points that have q as one of their k nearest neighbors. Existing methods for processing such queries have at least one of the following deficiencies: (i) they do not support arbitrary values of k (ii) they cannot deal efficiently with database updates, (iii) they are applicable only to 2D data (but not to higher dimensionality), and (iv) they retrieve only approximate results. Motivated by these shortcomings, we develop algorithms for exact processing of RkNN with arbitrary values of k on dynamic multidimensional datasets. Our methods utilize a conventional data-partitioning index on the dataset and do not require any pre-computation. In addition to their flexibility, we experimentally verify that the proposed algorithms outperform the existing ones even in their restricted focus. We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple ((s_i, _i) ) consisting of a point (s_i ) in a (d )-dimensional space and a probability ( _i ) determining its existence. The problem is interesting and non-trivial even in (1 )-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity ( (n^2) ). We then show that under more natural and less adversarial conditions, the size of the (1 )-dimensional LVD is significantly smaller: (1) ( (k n) ) if the input has only (k ) distinct probability values, (2) (O(n n) ) on average, and (3) (O(n n ) ) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries. In applications like location-based services, sensor monitoring and biological databases, the values of the database items are inherently uncertain in nature. An important query for uncertain objects is the probabilistic nearest-neighbor query (PNN), which computes the probability of each object for being the nearest neighbor of a query point. Evaluating this query is computationally expensive, since it needs to consider the relationship among uncertain objects, and requires the use of numerical integration or Monte-Carlo methods. Sometimes, a query user may not be concerned about the exact probability values. For example, he may only need answers that have sufficiently high confidence. We thus propose the constrained nearest-neighbor query (C-PNN), which returns the IDs of objects whose probabilities are higher than some threshold, with a given error bound in the answers. The C-PNN can be answered efficiently with probabilistic verifiers. These are methods that derive the lower and upper bounds of answer probabilities, so that an object can be quickly decided on whether it should be included in the answer. We have developed three probabilistic verifiers, which can be used on uncertain data with arbitrary probability density functions. Extensive experiments were performed to examine the effectiveness of these approaches.