### Abstract

This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n ^{4} log ^{O(1)} n) in an algebraic complexity model. We also achieve a time of O(n ^{4}L · μ(L)) in a bit complexity model where L is the maximum bit size of input numbers and μ(L) is the complexity of multiplying two L bit integers. These and several other results highlight a general linearization technique which transforms nonlinear problems into some higher-dimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo's parametric searching. We further report on experimental work comparing the practicality of an exact with that of a numerical strategy.

Original language | English (US) |
---|---|

Pages (from-to) | 170-186 |

Number of pages | 17 |

Journal | Algorithmica (New York) |

Volume | 27 |

Issue number | 2 |

State | Published - 2000 |

### Fingerprint

### Keywords

- ε-approximation algorithms
- Best-fit line
- Geometric optimization
- Parametric search
- Smallest enclosing cylinder

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Applied Mathematics
- Safety, Risk, Reliability and Quality

### Cite this

*Algorithmica (New York)*,

*27*(2), 170-186.

**Smallest enclosing cylinders.** / Schömer, E.; Sellen, J.; Teichmann, M.; Yap, Chee.

Research output: Contribution to journal › Article

*Algorithmica (New York)*, vol. 27, no. 2, pp. 170-186.

}

TY - JOUR

T1 - Smallest enclosing cylinders

AU - Schömer, E.

AU - Sellen, J.

AU - Teichmann, M.

AU - Yap, Chee

PY - 2000

Y1 - 2000

N2 - This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4L · μ(L)) in a bit complexity model where L is the maximum bit size of input numbers and μ(L) is the complexity of multiplying two L bit integers. These and several other results highlight a general linearization technique which transforms nonlinear problems into some higher-dimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo's parametric searching. We further report on experimental work comparing the practicality of an exact with that of a numerical strategy.

AB - This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4L · μ(L)) in a bit complexity model where L is the maximum bit size of input numbers and μ(L) is the complexity of multiplying two L bit integers. These and several other results highlight a general linearization technique which transforms nonlinear problems into some higher-dimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo's parametric searching. We further report on experimental work comparing the practicality of an exact with that of a numerical strategy.

KW - ε-approximation algorithms

KW - Best-fit line

KW - Geometric optimization

KW - Parametric search

KW - Smallest enclosing cylinder

UR - http://www.scopus.com/inward/record.url?scp=0000023717&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000023717&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 170

EP - 186

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2

ER -