Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets

Yu, Hai; Agarwal, Pankaj K.; Poreddy, Raghunath; Varadarajan, Kasturi

doi:10.1007/s00453-007-9067-9

Cited by 23 publications

(23 citation statements)

References 36 publications

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“…The notion of ε-kernels was introduced by Agarwal et al [7] and efficient algorithms for computing an ε-kernel of a set of n points in R d were given in [7,14,24]. Yu et al [24] also gave a simple and fast incremental algorithm for fitting various shapes through a given set of points.…”

Section: Related Resultsmentioning

confidence: 99%

“…Yu et al [24] also gave a simple and fast incremental algorithm for fitting various shapes through a given set of points. See [8] for a review of known results on coresets.…”

Section: Related Resultsmentioning

confidence: 99%

“…3 we present an incremental algorithm for shape fitting with k outliers, which is an extension of the incremental algorithm by Yu et al [24]. The algorithm works by repeatedly grating points from the original point set into a working set; the points that violate the current solution for the working set the most are selected by the algorithm.…”

Section: Related Resultsmentioning

confidence: 99%

“…At the beginning of the ith iteration, for 0 ≤ i ≤ 2k, we have a set P i ⊆ P ; initially P 0 = P . We compute a δ-kernel T i of P i , using an existing algorithm for computing δ-kernels [7,14,24]. We set P i+1 = P i \ T i .…”

Section: Algorithmmentioning

confidence: 99%

“…Roots of Polynomials To compute (k, ε)-kernels of fractional powers of polynomials, we need the following observation from [24] (see also [7]):…”

Section: Theorem 24 Given a One-pass Algorithm For Computing ε-Kernementioning

confidence: 99%

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Robust Shape Fitting via Peeling and Grating Coresets

Agarwal

Har-Peled

2007

Discrete Comput Geom

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Let P be a set of n points in R d . A subset S of P is called a (k, ε)-kernel if for every direction, the directional width of S ε-approximates that of P , when k "outliers" can be ignored in that direction. We show that a (k, ε)-kernel of P of size. The new algorithm works by repeatedly "peeling" away (0, ε)-kernels from the point set.We also present a simple ε-approximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly "grating" critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, near-linear ε-approximation algorithm for shape fitting with outliers in low dimensions.We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems.

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Section: Related Resultsmentioning

confidence: 99%

“…Yu et al [24] also gave a simple and fast incremental algorithm for fitting various shapes through a given set of points. See [8] for a review of known results on coresets.…”

Section: Related Resultsmentioning

confidence: 99%

Section: Related Resultsmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

“…Roots of Polynomials To compute (k, ε)-kernels of fractional powers of polynomials, we need the following observation from [24] (see also [7]):…”

Section: Theorem 24 Given a One-pass Algorithm For Computing ε-Kernementioning

confidence: 99%

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Robust Shape Fitting via Peeling and Grating Coresets

Agarwal

Har-Peled

2007

Discrete Comput Geom

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Shape Fitting on Point Sets with Probability Distributions

Löffler

Phillips

2009

Lecture Notes in Computer Science

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A typical computational geometry problem begins: Consider a set P of n points in R d . However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if we do not know the set P exactly, but rather we have a probability distribution µ p governing the location of each point p ∈ P ?Consider a set of (non-fixed) points P , and let µ P be the probability distribution of this set. We study several measures (e.g. the radius of the smallest enclosing ball, or the area of the smallest enclosing box) with respect to µ P . The solutions to these problems do not, as in the traditional case, consist of a single answer, but rather a distribution of answers. We hence describe a data structure, called an ε-quantization, that can approximate such a distribution within ε in O(1/ε) space. We also extend this data structure to answer higher dimensional queries of µ P (e.g. the length and width of the smallest enclosing box in R 2 ). Rather than compute a new data structure for each measure we are interested in, we can also compute a single data structure that allows us to answer many questions at once. This data structure, an (ε, α)-kernel, is based on α-kernel coresets and can be used to create approximate ε-quantizations for geometric problems involving extent measures.Thirdly, we introduce a data structure that can answer questions of the type 'what is the probability that point q is in the smallest enclosing ball of P ?' For a given distribution µ P and summarizing shape (e.g. the smallest enclosing ball), we define an ε-shape inclusion probability function to be a function that assigns to a query point q ∈ R d a value that is at most ε away from the probability that q is contained in this summarizing shape of P . This results in a probability description more directly linked to the space that the input points live in.We provide simple and efficient randomized algorithms for computing all of these data structures, which are easy to implement and practical. We provide some experimental results to assert this. We also provide more involved deterministic algorithms for ε-quantizations for problems involving shapes with bounded VC-dimension that run in time polynomial in n and 1/ε.

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Stability of ε-Kernels

Agarwal

Phillips

2010

Algorithms – ESA 2010

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Given a set P of n points in R d , an ε-kernel K ⊆ P approximates the directional width of P in every direction within a relative (1 − ε) factor. In this paper we study the stability of ε-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor ε. In the first case, we say an algorithm for dynamically maintaining a ε-kernel is stable if at most O(1) points change in K as one point is inserted or deleted from P . We describe an algorithm to maintain an ε-kernel of size O(1/ε (d−1)/2 ) in O(1/ε (d−1)/2 + log n) time per update. Not only does our algorithm maintain a stable ε-kernel, its update time is faster than any known algorithm that maintains an ε-kernel of size O(1/ε (d−1)/2 ). Next, we show that if there is an ε-kernel of P of size κ, which may be dramatically less than O(1/ε (d−1)/2 ), then there is an (ε/2)-kernel of P of size O(min{1/ε (d−1)/2 , κ d/2 log d−2 (1/ε)}). Moreover, there exists a point set P in R d and a parameter ε > 0 such that if every ε-kernel of P has size at least κ, then any (ε/2)-kernel of P has size Ω(κ d/2 ). 1

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Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets

Cited by 23 publications

References 36 publications

Robust Shape Fitting via Peeling and Grating Coresets

Robust Shape Fitting via Peeling and Grating Coresets

Shape Fitting on Point Sets with Probability Distributions

Stability of ε-Kernels

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