### Abstract

We present an efficient algorithm for solving the “smallest kenclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present several algorithms that run in O(nk log^{c} n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nklog^{2} n). When only O(n log n) storage is allowed, the running time is (formula presentd). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.

Original language | English (US) |
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Title of host publication | Algorithms and Data Structures - 3rd Workshop, WADS 1993, Proceedings |

Publisher | Springer Verlag |

Pages | 325-336 |

Number of pages | 12 |

Volume | 709 LNCS |

ISBN (Print) | 9783540571551 |

State | Published - 1993 |

Externally published | Yes |

Event | 3rd Workshop on Algorithms and Data Structures, WADS 1993 - Montreal, Canada Duration: Aug 11 1993 → Aug 13 1993 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 709 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 3rd Workshop on Algorithms and Data Structures, WADS 1993 |
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Country | Canada |

City | Montreal |

Period | 8/11/93 → 8/13/93 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 3rd Workshop, WADS 1993, Proceedings*(Vol. 709 LNCS, pp. 325-336). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 709 LNCS). Springer Verlag.

**Computing the smallest k-enclosing circle and related problems.** / Efrat, Alon; Sharir, Micha; Ziv, Alon.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Data Structures - 3rd Workshop, WADS 1993, Proceedings.*vol. 709 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 709 LNCS, Springer Verlag, pp. 325-336, 3rd Workshop on Algorithms and Data Structures, WADS 1993, Montreal, Canada, 8/11/93.

}

TY - GEN

T1 - Computing the smallest k-enclosing circle and related problems

AU - Efrat, Alon

AU - Sharir, Micha

AU - Ziv, Alon

PY - 1993

Y1 - 1993

N2 - We present an efficient algorithm for solving the “smallest kenclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present several algorithms that run in O(nk logc n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nklog2 n). When only O(n log n) storage is allowed, the running time is (formula presentd). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.

AB - We present an efficient algorithm for solving the “smallest kenclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present several algorithms that run in O(nk logc n) time, where the constant c depends on the storage that the algorithm is allowed. When using O(nk) storage, the problem can be solved in time O(nklog2 n). When only O(n log n) storage is allowed, the running time is (formula presentd). The method we describe can be easily extended to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P, which contains k points from a given planar set, and finding the smallest hypodrome of a given length and orientation (formally defined in Section 4) containing k points from a given planar set.

UR - http://www.scopus.com/inward/record.url?scp=85029531691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029531691&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783540571551

VL - 709 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 325

EP - 336

BT - Algorithms and Data Structures - 3rd Workshop, WADS 1993, Proceedings

PB - Springer Verlag

ER -