A space-optimal data-stream algorithm for coresets in the plane

Agarwal, Pankaj K.; Yu, Hai

doi:10.1145/1247069.1247071

Cited by 17 publications

(31 citation statements)

References 23 publications

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“…In fact, the notion behind coresets was introduced in the context of these problems in the seminal paper by Agarwal et al [4]. Since then, a number of papers have progressively reduced the space bound required to maintain coresets [8,23,24,99] with Arya and Chan giving an algorithm that stores O( −(d−1)∕2 ) points [14] in d-dimensional Euclidean space. We will briefly outline in Sect.…”

Section: Theorem 1 For Any Set Of N Points a Euclidean Space There Ementioning

confidence: 99%

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

Munteanu

Schwiegelshohn

2017

Künstl Intell

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Section: Theorem 1 For Any Set Of N Points a Euclidean Space There Ementioning

confidence: 99%

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

Munteanu

Schwiegelshohn

2017

Künstl Intell

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“…Assume that we know the approximated values of err vis (q i q j ) for 0 ≤ i ≤ j ≤ k + 1 by induction. Also, assume that we have k + 1 core-sets according to the algorithm of Agarwal and Yu [2] for point sets P (q i , p n ) where 0 ≤ i ≤ k from which we can find the extreme points in the direction of perpendicular to q i p n . When the new point p n+1 is given, it is added to these core-sets and a new core-set is created for segment q k+1 p n+1 .…”

Section: Lemma 8 Over the Visibility Polygon Of A Point Observer Thementioning

confidence: 99%

Efficient Observer-Dependent Simplification in Polygonal Domains

Zarei

Ghodsi

2011

Algorithmica

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In this paper, we consider a special version of the well-known linesimplification problem for simplifying the boundary of a region illuminated by a point light source q, or its visibility polygon VP(q). In this simplification approach, we should take the position of q as an essential factor into account to determine the quality of the resulting simplification. For this purpose, we redefine the known distanceand area-distortion error criteria as the main simplification criteria to take into account the distance between the observer q and the boundary of VP(q). Based on this, we propose algorithms for simplifying VP(q). More precisely, we propose simplification algorithms of O(n 2 ) and O(n 4/3+δ ) running time for observer-dependent distance-distortion simplification criterion and an O(n 3 ) simplification algorithm for observer-dependent area-distortion criterion where n is the number of vertices in VP(q). Moreover, we consider the observer-dependent distance-distortion simplification problem in the data streaming model where the vertices of VP(q) are given as a stream and only a constant amount of memory is available.

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“…Agarwal and Yu [2] have described a streaming algorithm for maintaining a coreset that can be used to approximate the width of a set in any direction. More precisely, they maintain an ε-coreset of size O(1/ √ ε) in O(log (1/ε)) amortized time per insertion.…”

Section: Let W(i J ) Be the Width Of The Points In Subpath P (I J) mentioning

confidence: 99%

“…We study line simplification in a streaming setting, where we only have a limited amount of storage so that we cannot store all the points. A similar streaming model for geometric algorithms has been used by, e.g., Agarwal and Yu [2] and Zarrabi-Zadeh and Chan [19]. Also see Muthukrishnan's survey [18] on streaming algorithms.…”

Section: Introductionmentioning

confidence: 99%

Streaming Algorithms for Line Simplification

Abam

Berg

Hachenberger

et al. 2009

Discrete Comput Geom

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We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0 , p 1 , p 2 , . . . in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k Comput Geom (2010) 43: 497-515 points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k 2 ) additional storage.

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A space-optimal data-stream algorithm for coresets in the plane

Cited by 17 publications

References 23 publications

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

Efficient Observer-Dependent Simplification in Polygonal Domains

Streaming Algorithms for Line Simplification

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