Expected computations on color spanning sets

Li, Chao; Fan, Chenglin; Luo, Jun; Zhong, Fang; Zhu, Binhai

doi:10.1007/s10878-014-9764-7

Cited by 6 publications

(7 citation statements)

References 24 publications

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“…Li et al [12] gave a deterministic (2/ √ 3)-approximation algorithm in R 2 , which is based on (exactly) computing the expected diameter of the stochastic smallest enclosing ball. Although [12] only considered the case in R 2 , the algorithm can be naturally extended to compute a ( √ 2d/ √ d + 1)-approximation of the expected diameter of a SCH in R d . Nevertheless, the runtime of this algorithm grows exponentially as d increases, since computing the expected diameter of the stochastic smallest enclosing ball requires n Ω(d) time [9].…”

Section: Related Workmentioning

confidence: 99%

“…There have also been several papers concerning SCH [3,8,12,14,16]. We only summarize those that are strongly relevant to this paper.…”

Section: Related Workmentioning

confidence: 99%

“…We only summarize those that are strongly relevant to this paper. Li et al [12] studied the expected computation of some basic statistics of a SCH in R 2 , e.g., area, perimeter, diameter (their results for diameter are summarized below), etc. The results in [12] are presented in a slightly different uncertainty model, but most of the algorithms also work under existential uncertainty.…”

Section: Related Workmentioning

confidence: 99%

“…In recent years, there have been a considerable amount of work regarding geometric problems on stochastic datasets. Among them, the convex hull structure under uncertainty, known as stochastic convex hull (SCH), has received a lot of attention [3,12,14,16].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Related Workmentioning

confidence: 99%

“…There have also been several papers concerning SCH [3,8,12,14,16]. We only summarize those that are strongly relevant to this paper.…”

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Xue

Janardan

2017

Lecture Notes in Computer Science

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“…More recently, in 2014, Li et al [15] considered a set of n points in the plane colored with k colors, and studied, among other computation problems, the computation of the expected area or perimeter of the convex hull of a random sample of the points. Such random samples are obtained by picking for each color a point of that color uniformly at random.…”

Section: Related Workmentioning

confidence: 99%

Area and Perimeter of the Convex Hull of Stochastic Points

Pérez-Lantero

2016

The Computer Journal

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Given a set P of n points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset S of P . The random subset S is formed by drawing each point p of P independently with a given rational probability π p . For both measures of the convex hull, we show that it is #P-hard to compute the probability that the measure is at least a given bound w. For ε ∈ (0, 1), we provide an algorithm that runs in O(n 6 /ε) time and returns a value that is between the probability that the area is at least w, and the probability that the area is at least (1 − ε)w. For the perimeter, we show a similar algorithm running in O(n 6 /ε) time. Finally, given ε, δ ∈ (0, 1) and for any measure, we show an O(n log n+(n/ε 2 ) log(1/δ))-time Monte Carlo algorithm that returns a value that, with probability of success at least 1 − δ, differs at most ε from the probability that the measure is at least w.

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The 2-Mixed-Center Color Spanning Problem

Wang,

Xu,

et al. 2023

Lecture Notes in Computer Science

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Expected computations on color spanning sets

Cited by 6 publications

References 24 publications

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

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

Area and Perimeter of the Convex Hull of Stochastic Points

The 2-Mixed-Center Color Spanning Problem

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