## 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 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.

Original language | English (US) |
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Title of host publication | Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings |

Editors | Faith Ellen, Antonina Kolokolova, Jorg-Rudiger Sack |

Publisher | Springer Verlag |

Pages | 581-592 |

Number of pages | 12 |

ISBN (Print) | 9783319621265 |

DOIs | |

State | Published - 2017 |

Event | 15th International Symposium on Algorithms and Data Structures, WADS 2017 - St. John’s, Canada Duration: Jul 31 2017 → Aug 2 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10389 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th International Symposium on Algorithms and Data Structures, WADS 2017 |
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Country/Territory | Canada |

City | St. John’s |

Period | 7/31/17 → 8/2/17 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing AG 2017.

## Keywords

- Combinatorial complexity
- Diameter
- Expectation
- Uncertain data
- Width