Abstract
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 a 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 logn) 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.
Original language | English (US) |
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Pages (from-to) | 16-31 |
Number of pages | 16 |
Journal | Computational Geometry: Theory and Applications |
Volume | 82 |
DOIs | |
State | Published - Sep 2019 |
Bibliographical note
Funding Information:A preliminary version appeared at the 15th Algorithms and Data Structures Symposium, 2017. The research of Jie Xue is supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota .
Funding Information:
A preliminary version appeared at the 15th Algorithms and Data Structures Symposium, 2017. The research of Jie Xue is supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota.? A preliminary version appeared at the 15th Algorithms and Data Structures Symposium, 2017. The research of Jie Xue is supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota.
Funding Information:
A preliminary version appeared at the 15th Algorithms and Data Structures Symposium, 2017. The research of Jie Xue is supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota. A preliminary version appeared at the 15th Algorithms and Data Structures Symposium, 2017. The research of Jie Xue is supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota.
Publisher Copyright:
© 2019 Elsevier B.V.
Keywords
- Approximation algorithm
- Convex hull
- Expectation
- Uncertain data