On the expected diameter, width, and complexity of a stochastic convex-hull

Jie Xue; Yuan Li; Ravi Janardan

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

On the expected diameter, width, and complexity of a stochastic convex-hull

Jie Xue, Yuan Li, Ravi Janardan

Computer Science and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

3 Scopus citations

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)
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	https://doi.org/10.1007/978-3-319-62127-2_49
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)
Volume	10389 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	15th International Symposium on Algorithms and Data Structures, WADS 2017
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

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

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Xue, J., Li, Y., & Janardan, R. (2017). On the expected diameter, width, and complexity of a stochastic convex-hull. In F. Ellen, A. Kolokolova, & J.-R. Sack (Eds.), Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings (pp. 581-592). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10389 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62127-2_49

On the expected diameter, width, and complexity of a stochastic convex-hull. / Xue, Jie; Li, Yuan; Janardan, Ravi.
Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. ed. / Faith Ellen; Antonina Kolokolova; Jorg-Rudiger Sack. Springer Verlag, 2017. p. 581-592 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10389 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Xue, J, Li, Y & Janardan, R 2017, On the expected diameter, width, and complexity of a stochastic convex-hull. in F Ellen, A Kolokolova & J-R Sack (eds), Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10389 LNCS, Springer Verlag, pp. 581-592, 15th International Symposium on Algorithms and Data Structures, WADS 2017, St. John’s, Canada, 7/31/17. https://doi.org/10.1007/978-3-319-62127-2_49

Xue J, Li Y, Janardan R. On the expected diameter, width, and complexity of a stochastic convex-hull. In Ellen F, Kolokolova A, Sack JR, editors, Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. Springer Verlag. 2017. p. 581-592. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-62127-2_49

Xue, Jie ; Li, Yuan ; Janardan, Ravi. / On the expected diameter, width, and complexity of a stochastic convex-hull. Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings. editor / Faith Ellen ; Antonina Kolokolova ; Jorg-Rudiger Sack. Springer Verlag, 2017. pp. 581-592 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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N2 - We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in ℝd 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(nd+1 log n) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(nd)-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

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