On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

Xue, Jie; Li, Yuan; Janardan, Ravi

doi:10.1007/978-3-319-62127-2_49

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…Some algorithms were deterministic, some of those based on a stochastic approach, e.g. Xue [ 27 ]. Efficient CH algorithms get more and more complex as far as implementation aspects are concerned.…”

Section: Finding a Diameter Of A Convex Hullmentioning

confidence: 99%

Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3

Skala

2020

Computational Science and Its Applications – ICCSA 2020

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Convex hull of points and its diameter computation is a frequent task in many engineering problems, However, in engineering solutions, the asymptotic computational complexity is less important than the computational complexity for the expected data size to be processed. This contribution describes “an engineering solution" of the convex hulls and their diameter computation using space-subdivision and data-reduction approaches. This approach proved a significant speed-up of computation with simplicity of implementation. Surprisingly, the experiments proved, that in the case of the space subdivision the reduction of points is so efficient, that the “brute force" algorithms for the convex hull and its diameter computation of the remaining points have nearly no influence to the time of computation.

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Section: Finding a Diameter Of A Convex Hullmentioning

confidence: 99%

Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3

Skala

2020

Computational Science and Its Applications – ICCSA 2020

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“…Cabello and Chan also drew connections between computing Shapley values and stochastic computational geometry models. There have been many studies on the behavior of the convex hull under unipoint model where each point has an existential probability, for example see [1,10,15,19,26,28]. In particular, Xue et al [28] discussed the expected diameter and width of the convex hull.…”

Section: Related Workmentioning

confidence: 99%

“…There have been many studies on the behavior of the convex hull under unipoint model where each point has an existential probability, for example see [1,10,15,19,26,28]. In particular, Xue et al [28] discussed the expected diameter and width of the convex hull. Huang et al [15] presented a way to construct ǫ-coreset for directional width under the model.…”

Section: Related Workmentioning

confidence: 99%

Computing Shapley Values for Mean Width in 3-D

Tan¹

2020

Preprint

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The Shapley value is a common tool in game theory to evaluate the importance of a player in a cooperative setting. In a geometric context, it provides a way to measure the contribution of a geometric object in a set towards some function on the set. Recently, Cabello and Chan (SoCG 2019) presented algorithms for computing Shapley values for a number of functions for point sets in the plane. More formally, a coalition game consists of a set of players N and a characteristic function v : 2 N → R with v(∅) = 0. Let π be a uniformly random permutation of N , and PN (π, i) be the set of players in N that appear before player i in the permutation π. The Shapley value of the game is defined to be φMore intuitively, the Shapley value represents the impact of player i's appearance over all insertion orders. We present an algorithm to compute Shapley values in 3-D, where we treat points as players and use the mean width of the convex hull as the characteristic function. Our algorithm runs in O(n 3 log 2 n) time and O(n) space. Our approach is based on a new data structure for a variant of the dynamic convolution problem (u, v, p), where we want to answer u • v dynamically. Our data structure supports updating u at position p, incrementing and decrementing p and rotating v by 1. We present a data structure that supports n operations in O(n log 2 n) time and O(n) space. Moreover, the same approach can be used to compute the Shapley values for the mean volume of the convex hull projection onto a uniformly random ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Data structures design and analysis Keywords and phrases Shapley value, mean width, dynamic convolutionAcknowledgements The author is very grateful to David Mount for discussions and advice for the paper. The author is also thankful to Geng Lin for proofreading and helping improving the readability. Computing Shapley Values for Mean Width in 3-DN that appear before player i in the permutation π. The Shapley value of player i ∈ N is defined to be(1)Intuitively, the Shapley value represents the expected marginal contribution of i to the objective function over all permutations of N . There are wide applications of Shapley values.A survey by Winter [27] and a book dedicated to this topic [24] provide insights on how the concept can be interpreted and applied in multiple ways, such as utility of players, allocation of resources of the grand coalition and measure of power in a voting system. Moreover, the values can be characterized axiomatically, making it the only natural quantity that satisfies certain properties. More details can be found in standard game theory textbooks ([14], [8, Chapter 5], [22, Section 9.4]).In convex geometry and measure theory, intrinsic volumes are a key concept to characterize the "size" and "shape" of a convex body regardless of the translation and rotation in its underlying space. For example, Steiner's formula [12] relates intrinsic volumes to the volume of the Minkowski-sum of a convex body ...

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On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

Xue

Janardan

2017

Lecture Notes in Computer Science

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We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in R d each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both n and d. For width, two approximation algorithms are provided: a deterministic O(1)-approximation running in O(n d+1 log n) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(n d )-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.

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On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

Cited by 5 publications

References 19 publications

Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3

Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3

Computing Shapley Values for Mean Width in 3-D

On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

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