Largest Bounding Box, Smallest Diameter, and Related Problems on Imprecise Points

Löffler, Maarten; Kreveld, Marc van

doi:10.1007/978-3-540-73951-7_39

Cited by 24 publications

(22 citation statements)

References 17 publications

(23 reference statements)

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“…In deterministic models, each point is assumed to be inside a given region (see e.g. [9,10,29,37]). Probabilistic models can be further classified into the existential model and the locational model.…”

Section: Previous Resultsmentioning

confidence: 99%

Nearest-neighbor searching under uncertainty

Agarwal

Aronov

Har-Peled

et al. 2012

Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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Nearest-neighbor queries, which ask for returning the nearest neighbor of a query point in a set of points, are important and widely studied in many fields because of a wide range of applications. In many of these applications, 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 and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance, which we refer to as the expected nearest neighbor (ENN). We present methods for computing an exact ENN or an ε-approximate ENN, for a given error parameter 0 < ε < 1, under different distance functions. These methods build an index of near-linear size and answer ENN queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or ε-approximate ENN queries with provable performance guarantees.

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Section: Previous Resultsmentioning

confidence: 99%

Nearest-neighbor searching under uncertainty

Agarwal

Aronov

Har-Peled

et al. 2012

Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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“…In a series of works [10,12,33,39], optimizing the overlap of two objects (both in 2D and 3D) under a rigid motion was studied. Löffler and van Kreveld [43,60] considered moving points to optimize certain measures of a point set. Martin and Stephenson [44] developed algorithms for testing (both in 2D and 3D) whether an object fits into a box (we use their result in one of our algorithms).…”

Section: Related Workmentioning

confidence: 99%

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

et al. 2013

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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.

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“…Khanban and co-authors [13,15] developed a theory for returning partial Delaunay or Voronoi diagrams, consisting of the portion of the diagram that is certain. Van Kreveld and Löffler [18,17] consider the problem of determining the smallest and largest possible values for geometric extent measures-such as the diameter or convex hull area-of a set of imprecise points.…”

Section: Related Workmentioning

confidence: 99%

Preprocessing Imprecise Points and Splitting Triangulations

Kreveld¹,

Löffler²,

Mitchell³

2010

SIAM J. Comput.

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Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.

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Largest Bounding Box, Smallest Diameter, and Related Problems on Imprecise Points

Cited by 24 publications

References 17 publications

Nearest-neighbor searching under uncertainty

Nearest-neighbor searching under uncertainty

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

Preprocessing Imprecise Points and Splitting Triangulations

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