TY - GEN

T1 - On the expected diameter, width, and complexity of a stochastic convex-hull

AU - Xue, Jie

AU - Li, Yuan

AU - Janardan, Ravi

N1 - Publisher Copyright:
© Springer International Publishing AG 2017.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017

Y1 - 2017

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.

AB - 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.

KW - Combinatorial complexity

KW - Diameter

KW - Expectation

KW - Uncertain data

KW - Width

UR - http://www.scopus.com/inward/record.url?scp=85025164636&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85025164636&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-62127-2_49

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

M3 - Conference contribution

AN - SCOPUS:85025164636

SN - 9783319621265

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 581

EP - 592

BT - Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings

A2 - Ellen, Faith

A2 - Kolokolova, Antonina

A2 - Sack, Jorg-Rudiger

PB - Springer Verlag

T2 - 15th International Symposium on Algorithms and Data Structures, WADS 2017

Y2 - 31 July 2017 through 2 August 2017

ER -