### Abstract

In this paper it is shown that the diameter D(P) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D(P) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD(P) intersects its corresponding edge in the FPVD(P) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.

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

Pages (from-to) | 43-61 |

Number of pages | 19 |

Journal | Machine Intelligence and Pattern Recognition |

Volume | 2 |

Issue number | C |

DOIs | |

State | Published - Jan 1 1985 |

### Keywords

- 3.36
- 3.63
- 5.25
- 5.30
- 5.5
- algorithms
- all-furthest-neighbor problem
- computational geometry
- convex hull
- diameter
- maximal spanning tree
- minimum spanning circle
- Voronoi diagram

### ASJC Scopus subject areas

- Computer Vision and Pattern Recognition
- Artificial Intelligence

### Cite this

*Machine Intelligence and Pattern Recognition*,

*2*(C), 43-61. https://doi.org/10.1016/B978-0-444-87806-9.50008-6

**On Geometric Algorithms that use the Furthest-Point Voronoi Diagram.** / Bhattacharya, Binay K.; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Machine Intelligence and Pattern Recognition*, vol. 2, no. C, pp. 43-61. https://doi.org/10.1016/B978-0-444-87806-9.50008-6

}

TY - JOUR

T1 - On Geometric Algorithms that use the Furthest-Point Voronoi Diagram

AU - Bhattacharya, Binay K.

AU - Toussaint, Godfried

PY - 1985/1/1

Y1 - 1985/1/1

N2 - In this paper it is shown that the diameter D(P) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D(P) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD(P) intersects its corresponding edge in the FPVD(P) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.

AB - In this paper it is shown that the diameter D(P) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in [1] and [2]. It is also proved that if P is contained in the disk determined by D(P) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD(P) intersects its corresponding edge in the FPVD(P) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in [1] and [2]. A proof of correctness is given for the minimum spanning circle algorithm proposed in [2] and [3]. Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.

KW - 3.36

KW - 3.63

KW - 5.25

KW - 5.30

KW - 5.5

KW - algorithms

KW - all-furthest-neighbor problem

KW - computational geometry

KW - convex hull

KW - diameter

KW - maximal spanning tree

KW - minimum spanning circle

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=85012589136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012589136&partnerID=8YFLogxK

U2 - 10.1016/B978-0-444-87806-9.50008-6

DO - 10.1016/B978-0-444-87806-9.50008-6

M3 - Article

VL - 2

SP - 43

EP - 61

JO - Machine Intelligence and Pattern Recognition

JF - Machine Intelligence and Pattern Recognition

SN - 0923-0459

IS - C

ER -