### Abstract

Consider n i.i.d. random lines in the plane defined by their slope and distance from the origin. The slope is uniformly distributed on [0, 2π] and independent of the distance R from the origin. These lines define a set I of n(n - 1)/2 intersection points. It was recently shown by Atallah and Ching and Lee that the cardinality of the convex hull of these intersection points is O(n), and they exhibited an O(n log n) time algorithm for computing such a convex hull. Let N_{ch} and N_{ol} be the number of points on the convex hull and outer layer of I, respectively. We show that there exist arrangements of lines in which N_{ol} = n(n - l)/2. We show that, nevertheless, both N_{ch} and N_{ol} have expected values O(1), and give bounds that are uniform over all distributions of R with 0 ≤ ER < ∞. These results lead to an algorithm for computing the convex hull of I in O(n log n) worst-case time and O(n) expected time under these conditions.

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

Pages (from-to) | 381-394 |

Number of pages | 14 |

Journal | Journal of Algorithms |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - May 1 1993 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics

### Cite this

*Journal of Algorithms*,

*14*(3), 381-394. https://doi.org/10.1006/jagm.1993.1020

**Convex Hulls for Random Lines.** / Devroye, L.; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Journal of Algorithms*, vol. 14, no. 3, pp. 381-394. https://doi.org/10.1006/jagm.1993.1020

}

TY - JOUR

T1 - Convex Hulls for Random Lines

AU - Devroye, L.

AU - Toussaint, Godfried

PY - 1993/5/1

Y1 - 1993/5/1

N2 - Consider n i.i.d. random lines in the plane defined by their slope and distance from the origin. The slope is uniformly distributed on [0, 2π] and independent of the distance R from the origin. These lines define a set I of n(n - 1)/2 intersection points. It was recently shown by Atallah and Ching and Lee that the cardinality of the convex hull of these intersection points is O(n), and they exhibited an O(n log n) time algorithm for computing such a convex hull. Let Nch and Nol be the number of points on the convex hull and outer layer of I, respectively. We show that there exist arrangements of lines in which Nol = n(n - l)/2. We show that, nevertheless, both Nch and Nol have expected values O(1), and give bounds that are uniform over all distributions of R with 0 ≤ ER < ∞. These results lead to an algorithm for computing the convex hull of I in O(n log n) worst-case time and O(n) expected time under these conditions.

AB - Consider n i.i.d. random lines in the plane defined by their slope and distance from the origin. The slope is uniformly distributed on [0, 2π] and independent of the distance R from the origin. These lines define a set I of n(n - 1)/2 intersection points. It was recently shown by Atallah and Ching and Lee that the cardinality of the convex hull of these intersection points is O(n), and they exhibited an O(n log n) time algorithm for computing such a convex hull. Let Nch and Nol be the number of points on the convex hull and outer layer of I, respectively. We show that there exist arrangements of lines in which Nol = n(n - l)/2. We show that, nevertheless, both Nch and Nol have expected values O(1), and give bounds that are uniform over all distributions of R with 0 ≤ ER < ∞. These results lead to an algorithm for computing the convex hull of I in O(n log n) worst-case time and O(n) expected time under these conditions.

UR - http://www.scopus.com/inward/record.url?scp=0141947722&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141947722&partnerID=8YFLogxK

U2 - 10.1006/jagm.1993.1020

DO - 10.1006/jagm.1993.1020

M3 - Article

VL - 14

SP - 381

EP - 394

JO - Journal of Algorithms

JF - Journal of Algorithms

SN - 0196-6774

IS - 3

ER -