### Abstract

We present an efficient algorithm for solving the "smallest k-enclosing 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 two solutions. When using O(nk) storage, the problem can be solved in time O(nk log^{2} n). When only O(n log n) storage is allowed, the running time is O(nk log^{2} n log n/k). We also extend our technique 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 disk intersecting k segments from a given planar set of non-intersecting segments.

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

Pages (from-to) | 119-136 |

Number of pages | 18 |

Journal | Computational Geometry: Theory and Applications |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - 1994 |

Externally published | Yes |

### Fingerprint

### Keywords

- Geometric optimization
- Smallest enclosing circle

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*4*(3), 119-136. https://doi.org/10.1016/0925-7721(94)90003-5

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

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 4, no. 3, pp. 119-136. https://doi.org/10.1016/0925-7721(94)90003-5

}

TY - JOUR

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

AU - Efrat, Alon

AU - Sharir, Micha

AU - Ziv, Alon

PY - 1994

Y1 - 1994

N2 - We present an efficient algorithm for solving the "smallest k-enclosing 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 two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique 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 disk intersecting k segments from a given planar set of non-intersecting segments.

AB - We present an efficient algorithm for solving the "smallest k-enclosing 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 two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique 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 disk intersecting k segments from a given planar set of non-intersecting segments.

KW - Geometric optimization

KW - Smallest enclosing circle

UR - http://www.scopus.com/inward/record.url?scp=38149145977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149145977&partnerID=8YFLogxK

U2 - 10.1016/0925-7721(94)90003-5

DO - 10.1016/0925-7721(94)90003-5

M3 - Article

VL - 4

SP - 119

EP - 136

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 3

ER -