We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of @math points in @math each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both @math and @math . For width, two approximation algorithms are provided: a deterministic @math -approximation running in @math time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact @math -time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.
Geometric computation on uncertain data has received considerable attentions in recent years. A general introduction can be found in @cite_12 . Many fundamental geometric problems have been studied under uncertainty, e.g., nearest-neighbor search @cite_4 @cite_14 , minimum spanning trees @cite_9 , closest pair @cite_17 @cite_7 @cite_10 , range search @cite_18 @cite_15 , linear separability @cite_2 @cite_16 , dominance relation @cite_8 , etc. These problems were studied either under existential uncertainty (which is used in this paper) or under locational uncertainty (where the locations of the data points are uncertain).
@cite_18 @cite_14 @cite_4 @cite_7 @cite_8 @cite_9 @cite_2 @cite_15 @cite_16 @cite_10 @cite_12 @cite_17
1514180083 2963075731 2345659331 2147717514 1502916507 2124120807 1672197616 2401996954 1968623080 2045622468 2950981509 1547243772
We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, (k )-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be ( # P )-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge. With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an epsilon-EXP-KERNEL), as well as the probability distribution on the width (an (epsilon, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(epsilon^ -(d-1) 2 ) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an epsilon-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (epsilon, tau)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty. Uncertain data has been very common in many applications. In this paper, we consider the one-center problem for uncertain data on tree networks. In this problem, we are given a tree T and n (weighted) uncertain points each of which has m possible locations on T associated with probabilities. The goal is to find a point @math xź on T such that the maximum (weighted) expected distance from @math xź to all uncertain points is minimized. To the best of our knowledge, this problem has not been studied before. We propose a refined prune-and-search technique that solves the problem in linear time. We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). A fundamental and well-studied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S^'@?S that is contained in a query range (e.g., an axes-parallel rectangle) can be reported efficiently. However, in many situations, what is of interest is to generate a more informative ''summary'' of the output, obtained by applying a suitable aggregation function on S^'. Examples of such aggregation functions include count, sum, min, max, mean, median, mode, and top-k that are usually computed on a set of weights defined suitably on the objects. Such range-aggregate query problems have been the subject of much recent research in both the database and the computational geometry communities. In this paper, we further generalize this line of work by considering aggregation functions on point-sets that measure the extent or ''spread'' of the objects in the retrieved set S^'. The functions considered here include closest pair, diameter, and width. The challenge here is that these aggregation functions (unlike, say, count) are not efficiently decomposable in the sense that the answer to S^' cannot be inferred easily from answers to subsets that induce a partition of S^'. Nevertheless, we have been able to obtain space- and query-time-efficient solutions to several such problems including: closest pair queries with axes-parallel rectangles on point sets in the plane and on random point-sets in R^d (d>=2), closest pair queries with disks on random point-sets in the plane, diameter queries on point-sets in the plane, and guaranteed-quality approximations for diameter and width queries in the plane. Our results are based on a combination of geometric techniques, including multilevel range trees, Voronoi Diagrams, Euclidean Minimum Spanning Trees, sparse representations of candidate outputs, and proofs of (expected) upper bounds on the sizes of such representations. We explore the effect of dimensionality on the "nearest neighbor" problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality! Uncertain data has been very common in many applications. In this paper, we consider the one-center problem for uncertain data on tree networks. In this problem, we are given a tree T and n (weighted) uncertain points each of which has m possible locations on T associated with probabilities. The goal is to find a point (x^* ) on T such that the maximum (weighted) expected distance from (x^* ) to all uncertain points is minimized. To the best of our knowledge, this problem has not been studied before. We propose a refined prune-and-search technique that solves the problem in linear time. Abstract Geometric matching under uncertainty is a long-standing problem in computer vision. This paper presents a simple and efficient branch-and-bound algorithm for finding globally optimal solutions to geometric matching problems under a wide variety of allowable transformations (translations, isometries, equiform transformations, others) and a wide variety of allowable feature types (point features, oriented point features, line features, line segment features, etc.). The algorithm only requires an implementation of the forward transformation (model-to-image) and an error model to be supplied. Benchmarks and comparisons of the algorithm in comparison with alignment and Hough transform methods are presented. Conventional spatial queries, such as range search and nearest neighbor retrieval, involve only conditions on objects' geometric properties. Today, many modern applications call for novel forms of queries that aim to find objects satisfying both a spatial predicate, and a predicate on their associated texts. For example, instead of considering all the restaurants, a nearest neighbor query would instead ask for the restaurant that is the closest among those whose menus contain “steak, spaghetti, brandy” all at the same time. Currently, the best solution to such queries is based on the IR 2-tree, which, as shown in this paper, has a few deficiencies that seriously impact its efficiency. Motivated by this, we develop a new access method called the spatial inverted index that extends the conventional inverted index to cope with multidimensional data, and comes with algorithms that can answer nearest neighbor queries with keywords in real time. As verified by experiments, the proposed techniques outperform the IR 2-tree in query response time significantly, often by a factor of orders of magnitude. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , 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 @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. We study computing with indecisive point sets. Such points have spatial uncertainty where the true location is one of a finite number of possible locations. This data arises from probing distributions a few times or when the location is one of a few locations from a known database. In particular, we study computing distributions of geometric functions such as the radius of the smallest enclosing ball and the diameter. Surprisingly, we can compute the distribution of the radius of the smallest enclosing ball exactly in polynomial time, but computing the same distribution for the diameter is #P-hard. We generalize our polynomial-time algorithm to all LP-type problems. We also utilize our indecisive framework to deterministically and approximately compute on a more general class of uncertain data where the location of each point is given by a probability distribution.