Preprocessing Imprecise Points and Splitting Triangulations

Kreveld, Marc van; Löffler, Maarten; Mitchell, Joseph S. B.

doi:10.1007/978-3-540-92182-0_49

Cited by 14 publications

(14 citation statements)

References 16 publications

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“…The framework for preprocessing regions that represent points was first introduced by Held and Mitchell [12], who show how to store a set of disjoint unit disks in a data structure such that any point set containing one point from each disk can be triangulated in linear time. This result was later extended to arbitrary disjoint regions in the plane by van Kreveld et al [17]. Löffler and Snoeyink first showed that the Delaunay triangulation (or its dual, the Voronoi diagram) can also be computed in linear time after preprocessing a set of disjoint unit disks [18].…”

Section: Related Workmentioning

confidence: 99%

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

Löffler

Mulzer

2013

Lecture Notes in Computer Science

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Abstract. Let D be a set of n pairwise disjoint unit disks in the plane. 8We describe how to build a data structure for D so that for any point 9 set P containing exactly one point from each disk, we can quickly find 10 the onion decomposition (convex layers) of P . 11Our data structure can be built in O(n log n) time and has linear size. 12Given P , we can find its onion decomposition in O(n log k) time, where 13 k is the number of layers. We also provide a matching lower bound. 14Our solution is based on a recursive space decomposition, combined with 15 a fast algorithm to compute the union of two disjoint onion decomposi- 16tions.

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Section: Related Workmentioning

confidence: 99%

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

Löffler

Mulzer

2013

Lecture Notes in Computer Science

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“…Finally, we mention the paradigm of preprocessing imprecise points [7,17,20,22,24]. Given a set R of planar regions, we must preprocess R to quickly find the (Delaunay) triangulation or convex hull for inputs with exactly one point from each region in R. If we consider inputs with a random point from each region, the self-improving setting applies, and the previous results bound the expected running time in the limiting phase.…”

Section: Main Theoremsmentioning

confidence: 99%

Self-improving algorithms for coordinate-wise maxima

Clarkson

Mulzer

Seshadhri

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry. We give an algorithm for this problem in the selfimproving setting. We have n (unknown) independent distributions D1, D2, . . . , Dn of planar points. An input pointset (p1, p2, . . . , pn) is generated by taking an independent sample pi from each Di, so the input distribution D is the product i Di. A self-improving algorithm repeatedly gets input sets from the distribution D (which is a priori unknown) and tries to optimize its running time for D. Our algorithm uses the first few inputs to learn salient features of the distribution, and then becomes an optimal algorithm for distribution D. Let OPTD denote the expected depth of an optimal linear comparison tree computing the maxima for distribution D. Our algorithm eventually has an expected running time of O(OPTD + n), even though it did not know D to begin with.Our result requires new tools to understand linear comparison trees for computing maxima. We show how to convert general linear comparison trees to very restricted versions, which can then be related to the running time of our algorithm. An interesting feature of our algorithm is an interleaved search, where the algorithm tries to determine the likeliest point to be maximal with minimal computation. This allows the running time to be truly optimal for the distribution D.

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“…Löffler and Snoeyink [22] were the first to interpret R as a collection of imprecise measurements of a true point set P . The size of Ξ and the running time of the reconstruction phase, together quantify the information about (the Delaunay triangulation of) P contained in R. This interpretation was widely adopted within computational geometry and motivated many recent results for constructing Delaunay triangulations [4,5,11,28], spanning trees [20,30], convex hulls [15,16,23,25] and other planar decompositions [21,27] for imprecise points.…”

Section: Introductionmentioning

confidence: 99%

Preprocessing Imprecise Points for the Pareto Front

Hoog¹,

Kostitsyna²,

Löffler³

et al. 2021

Preprint

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In the preprocessing model for uncertain data we are given a set of regions R which model the uncertainty associated with an unknown set of points P . In this model there are two phases: a preprocessing phase, in which we have access only to R, followed by a reconstruction phase, in which we have access to points in P at a certain retrieval cost C per point. We study the following algorithmic question: how fast can we construct the Pareto front of P in the preprocessing model?We show that if R is a set of pairwise-disjoint axis-aligned rectangles, then we can preprocess R to reconstruct the Pareto front of P efficiently. To refine our algorithmic analysis, we introduce a new notion of algorithmic optimality which relates to the entropy of the uncertainty regions. Our proposed uncertainty-region optimality falls on the spectrum between worst-case optimality and instance optimality. We prove that instance optimality is unobtainable in the preprocessing model, whenever the classic algorithmic problem reduces to sorting. Our results are worst-case optimal in the preprocessing phase; in the reconstruction phase, our results are uncertainty-region optimal with respect to real RAM instructions, and instance optimal with respect to point retrievals.

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Preprocessing Imprecise Points and Splitting Triangulations

Cited by 14 publications

References 16 publications

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

Self-improving algorithms for coordinate-wise maxima

Preprocessing Imprecise Points for the Pareto Front

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