### Abstract

Given a set P of n points on the plane, a symmetric furthest-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in P. A pair of points is antipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of P is both a set of extreme points of P and an antipodal pair of P. It is also shown that an asymmetric furthest-neighbor (ASFN) pair is not necessarily antipodal. Furthermore, if P is such that no two distances are equal, it is shown that as many as, and no more than, ⌊n/2⌋ pairs of points are SFN pairs. A polygon is unimodal if for each vertex p_{k}, k = 1,...,n the distance function defined by the euclidean distance between p_{k} and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require 0(n log n) running time. It is shown that the above results lead to an 0(n) algorithm for computing all the SFN pairs of vertices of a unimodal polygon.

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

Pages (from-to) | 747-754 |

Number of pages | 8 |

Journal | Computers and Mathematics with Applications |

Volume | 9 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1983 |

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### ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

**The symmetric all-furthest- neighbor problem.** / Toussaint, Godfried.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 9, no. 6, pp. 747-754. https://doi.org/10.1016/0898-1221(83)90139-6

}

TY - JOUR

T1 - The symmetric all-furthest- neighbor problem

AU - Toussaint, Godfried

PY - 1983/1/1

Y1 - 1983/1/1

N2 - Given a set P of n points on the plane, a symmetric furthest-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in P. A pair of points is antipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of P is both a set of extreme points of P and an antipodal pair of P. It is also shown that an asymmetric furthest-neighbor (ASFN) pair is not necessarily antipodal. Furthermore, if P is such that no two distances are equal, it is shown that as many as, and no more than, ⌊n/2⌋ pairs of points are SFN pairs. A polygon is unimodal if for each vertex pk, k = 1,...,n the distance function defined by the euclidean distance between pk and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require 0(n log n) running time. It is shown that the above results lead to an 0(n) algorithm for computing all the SFN pairs of vertices of a unimodal polygon.

AB - Given a set P of n points on the plane, a symmetric furthest-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in P. A pair of points is antipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of P is both a set of extreme points of P and an antipodal pair of P. It is also shown that an asymmetric furthest-neighbor (ASFN) pair is not necessarily antipodal. Furthermore, if P is such that no two distances are equal, it is shown that as many as, and no more than, ⌊n/2⌋ pairs of points are SFN pairs. A polygon is unimodal if for each vertex pk, k = 1,...,n the distance function defined by the euclidean distance between pk and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require 0(n log n) running time. It is shown that the above results lead to an 0(n) algorithm for computing all the SFN pairs of vertices of a unimodal polygon.

UR - http://www.scopus.com/inward/record.url?scp=0020880698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0020880698&partnerID=8YFLogxK

U2 - 10.1016/0898-1221(83)90139-6

DO - 10.1016/0898-1221(83)90139-6

M3 - Article

AN - SCOPUS:0020880698

VL - 9

SP - 747

EP - 754

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 6

ER -