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Approximate) uncertain skylines
 In ICDT
, 2011
"... Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point’s uncertainty is described as a probability distribution over a discrete ..."
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Cited by 2 (1 self)
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Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point’s uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution. We also suggest why we believe our solution might be optimal. Next, we describe simple, nearlinear time approximation algorithms for computing the probability of each point lying on the skyline. In addition, some of our methods can be adapted to construct data structures that can efficiently determine the probability of a query point lying on the skyline. 1.
Geometric Computations on Indecisive Points ∗
, 2011
"... 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 ..."
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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 #Phard. We generalize our polynomialtime algorithm to all LPtype 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.
Range Counting Coresets for Uncertain Data
, 2013
"... We study coresets for various types of range counting queries on uncertain data. In our model each uncertain point has a probability density describing its location, sometimes defined as k distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncer ..."
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We study coresets for various types of range counting queries on uncertain data. In our model each uncertain point has a probability density describing its location, sometimes defined as k distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by just examining this subset. We study three distinct types of queries. RE queries return the expected number of points in a query range. RC queries return the number of points in the range with probability at least a threshold. RQ queries returns the probability that fewer than some threshold fraction of the points are in the range. In both RC and RQ coresets the threshold is provided as part of the query. And for each type of query we provide coreset constructions with approximationsize tradeoffs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancybased approaches on axisaligned range queries.
Uncertainty Visualization in HARDI based on Ensembles of ODFs
"... Visualization of the uncertainty in two diffusion shapes. (a) Two fibers crossing at 60 degrees with relative weight of 0.6:0.4 and SNR of 10. (b) Two fibers crossing at 90 degrees with equal weight and SNR of 20 (with much less uncertainty). In this paper, we propose a new and accurate technique fo ..."
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Visualization of the uncertainty in two diffusion shapes. (a) Two fibers crossing at 60 degrees with relative weight of 0.6:0.4 and SNR of 10. (b) Two fibers crossing at 90 degrees with equal weight and SNR of 20 (with much less uncertainty). In this paper, we propose a new and accurate technique for uncertainty analysis and uncertainty visualization based on fiber orientation distribution function (ODF) glyphs, associated with high angular resolution diffusion imaging (HARDI). Our visualization applies volume rendering techniques to an ensemble of 3D ODF glyphs, which we call SIP functions of diffusion shapes, to capture their variability due to underlying uncertainty. This rendering elucidates the complex heteroscedastic structural variation in these shapes. Furthermore, we quantify the extent of this variation by measuring the fraction of the volume of these shapes, which is consistent across all noise levels, the certain volume ratio. Our uncertainty analysis and visualization framework is then applied to synthetic data, as well as to HARDI humanbrain data, to study the impact of various image acquisition parameters and background noise levels on the diffusion shapes. 1
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing ✩
"... Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in th ..."
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Motivated by the desire to cope with data imprecision [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log ∗ n) expected time. If we further assume that the points are “oblivious ” with respect to the data structure, the running time improves to O(nα(n)). The same result holds when L is a set of line segments (in general position). We present several extensions, including a tradeoff between space and query time and an outputsensitive algorithm. We also study the “dual problem ” where we show how to efficiently compute the ( ≤ k)level of n lines in the plane, each of which is incident to a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their xorder), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the “standard ” convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).