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

  • Jie XueEmail author
  • Yuan Li
  • Ravi Janardan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in \(\mathbb {R}^d\) 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.


Uncertain data Expectation Diameter Width Combinatorial complexity 


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Minnesota — Twin CitiesMinneapolisUSA

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