### Abstract

Let Q = {q_{1}, q_{2},..., q_{n}} be a set of n points on the plane. The largest empty circle (LEG) problem consists in finding the largest circle C with center in the convex hull of Q such that no point q_{i} εQ lies in the interior of C. Shamos recently outlined an O(n log n) algorithm for solving this problem.^{(9)} In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in an O(n log n) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center of C is constrained to lie in an arbitrary convex n-gon, an 0(n log n) algorithm can still be obtained. Finally, an 0(n log n +k log n) algorithm is given for solving this problem when the center of C is constrained to lie in an arbitrary simple n-gon P. where k denotes the number of intersections occurring between edges of P and edges of the Voronoi diagram of Q and k ≤O(n^{2}).

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

Pages (from-to) | 347-358 |

Number of pages | 12 |

Journal | International Journal of Computer & Information Sciences |

Volume | 12 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 1983 |

### Fingerprint

### Keywords

- algorithms
- complexity
- computational geometry
- facility location
- Largest empty circle
- operations research
- polygons
- Voronoi diagram

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Engineering(all)

### Cite this

**Computing largest empty circles with location constraints.** / Toussaint, Godfried.

Research output: Contribution to journal › Article

*International Journal of Computer & Information Sciences*, vol. 12, no. 5, pp. 347-358. https://doi.org/10.1007/BF01008046

}

TY - JOUR

T1 - Computing largest empty circles with location constraints

AU - Toussaint, Godfried

PY - 1983/10/1

Y1 - 1983/10/1

N2 - Let Q = {q1, q2,..., qn} be a set of n points on the plane. The largest empty circle (LEG) problem consists in finding the largest circle C with center in the convex hull of Q such that no point qi εQ lies in the interior of C. Shamos recently outlined an O(n log n) algorithm for solving this problem.(9) In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in an O(n log n) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center of C is constrained to lie in an arbitrary convex n-gon, an 0(n log n) algorithm can still be obtained. Finally, an 0(n log n +k log n) algorithm is given for solving this problem when the center of C is constrained to lie in an arbitrary simple n-gon P. where k denotes the number of intersections occurring between edges of P and edges of the Voronoi diagram of Q and k ≤O(n2).

AB - Let Q = {q1, q2,..., qn} be a set of n points on the plane. The largest empty circle (LEG) problem consists in finding the largest circle C with center in the convex hull of Q such that no point qi εQ lies in the interior of C. Shamos recently outlined an O(n log n) algorithm for solving this problem.(9) In this paper it is shown that this algorithm does not always work correctly. A different approach is proposed here and shown to also result in an O(n log n) algorithm. The new approach has the advantage that it can also solve more general problems. In particular, it is shown that if the center of C is constrained to lie in an arbitrary convex n-gon, an 0(n log n) algorithm can still be obtained. Finally, an 0(n log n +k log n) algorithm is given for solving this problem when the center of C is constrained to lie in an arbitrary simple n-gon P. where k denotes the number of intersections occurring between edges of P and edges of the Voronoi diagram of Q and k ≤O(n2).

KW - algorithms

KW - complexity

KW - computational geometry

KW - facility location

KW - Largest empty circle

KW - operations research

KW - polygons

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=0020829957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0020829957&partnerID=8YFLogxK

U2 - 10.1007/BF01008046

DO - 10.1007/BF01008046

M3 - Article

AN - SCOPUS:0020829957

VL - 12

SP - 347

EP - 358

JO - International Journal of Parallel Programming

JF - International Journal of Parallel Programming

SN - 0885-7458

IS - 5

ER -