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## Geometric Clusterings (1990)

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Citations: | 29 - 1 self |

### Citations

774 |
Algorithms in Combinatorial Geometry
- Edelsbrunner
- 1987
(Show Context)
Citation Context ...he plane by assigning every point to the disk Di for which its power d(x, Mi) 2 − r2 i is minimal, we get the so-called power diagram (cf. Aurenhammer [2]; Imai, Iri, and Murota [12]; or Edelsbrunne=-=r [7]-=-, section 13.6). It is known that the power diagram is a dissection of the plane into (at most) k convex polygonal regions, very much like in the case of Voronoi diagrams, which are a special case of ... |

237 | The NP-completeness column: an ongoing guide
- Johnson
- 1987
(Show Context)
Citation Context ...NP-complete to decide whether a set of points can be covered by a given number of lines. For more information, the interested reader is referred to Brucker [5] and to Johnson’s NP-Completeness Colum=-=n [13]. In-=- this note we show that for every fixed k, the geometric k-clustering problem becomes solvable in polynomial time, if W and F are as follows: • W is the diameter or the radius; •Fis an arbitrary m... |

188 |
Optimal algorithms for approximate clustering
- Feder, Greene
- 1988
(Show Context)
Citation Context ...mplete (Megiddo and Supowit [14]). It is even NP-hard to find a solution whose maximum radius (or maximum diameter) is within a factor of 1.82 (or 1.97, respectively) of the optimum (Feder and Greene =-=[9]-=-). For fixed k, a polynomial algorithm for minimizing the maximum radius has been given by Drezner [6]. NP-completeness can also be shown for minimizing the maximum cluster area and for minimizing the... |

140 | On the complexity of some common geometric location problems
- Megiddo, Supowit
- 1984
(Show Context)
Citation Context ...clustering is NP-complete. The related problem of minimizing the maximum radius, which in the area of location problems is also known as the k-center problem, is also NP-complete (Megiddo and Supowit =-=[14]-=-). It is even NP-hard to find a solution whose maximum radius (or maximum diameter) is within a factor of 1.82 (or 1.97, respectively) of the optimum (Feder and Greene [9]). For fixed k, a polynomial ... |

115 |
Power diagrams: Properties, algorithms and applications,
- Aurenhammer
- 1987
(Show Context)
Citation Context ...y pair of non-concentric disks. If we partition the plane by assigning every point to the disk Di for which its power d(x, Mi) 2 − r2 i is minimal, we get the so-called power diagram (cf. Aurenhamme=-=r [2]-=-; Imai, Iri, and Murota [12]; or Edelsbrunner [7], section 13.6). It is known that the power diagram is a dissection of the plane into (at most) k convex polygonal regions, very much like in the case ... |

64 |
On the complexity of clustering problems,
- Brucker
- 1978
(Show Context)
Citation Context ...result of Megiddo and Tamir [15] that it is NP-complete to decide whether a set of points can be covered by a given number of lines. For more information, the interested reader is referred to Brucker =-=[5] and-=- to Johnson’s NP-Completeness Column [13]. In this note we show that for every fixed k, the geometric k-clustering problem becomes solvable in polynomial time, if W and F are as follows: • W is th... |

44 |
Finding tailored partitions
- Hershberger, Suri
- 1991
(Show Context)
Citation Context ...e straightforward way. This algorithm is derived in section 3. The problem of testing whether a 2-clustering with specified bounds on the two diameters exists has been treated by Hershberger and Suri =-=[11]-=-. They gave an O(n log n)-time algorithm which does not use the separability of the two clusters. Our paper establishes the polynomial complexity status of a class of clustering problems, and it gives... |

42 |
Clustering Algorithms based on Minimum and Maximum Spanning Trees
- Asano, Bhattacharya, et al.
- 1988
(Show Context)
Citation Context ...g functions F are the maximum, the sum, or the sum of the squares of k non-negative arguments. The 2-clustering problem for the maximum diameter has been treated by Asano, Bhattacharya, Keil, and Yao =-=[1]-=-. They gave an O(n log n) algorithm for this problem. Monma and Suri [16] gave an O(n 2 ) algorithm for finding a 2-clustering with smallest sum of diameters. Some further related results are discusse... |

38 |
On the complexity of locating linear facilities in the plane
- Megiddo, Tamir
- 1982
(Show Context)
Citation Context ...has been given by Drezner [6]. NP-completeness can also be shown for minimizing the maximum cluster area and for minimizing the sum of all cluster areas, as follows from a result of Megiddo and Tamir =-=[15] t-=-hat it is NP-complete to decide whether a set of points can be covered by a given number of lines. For more information, the interested reader is referred to Brucker [5] and to Johnson’s NP-Complete... |

27 |
The p-center problem: heuristic and optimal algorithms,
- Drezner
- 1984
(Show Context)
Citation Context ...mum diameter) is within a factor of 1.82 (or 1.97, respectively) of the optimum (Feder and Greene [9]). For fixed k, a polynomial algorithm for minimizing the maximum radius has been given by Drezner =-=[6]-=-. NP-completeness can also be shown for minimizing the maximum cluster area and for minimizing the sum of all cluster areas, as follows from a result of Megiddo and Tamir [15] that it is NP-complete t... |

22 |
On clustering problems with connected optima
- Boros, Hammer
- 1989
(Show Context)
Citation Context ...uster centers. Similarly, for the problem where the sum of the squares of all distances between points in the same cluster is to be minimized (without division by the cluster sizes), Boros and Hammer =-=[3]-=- showed that two clusters in an optimal solution can always be separated by circle (or a sphere, in higher dimensions). In both of these cases the separability result is due to the special form of the... |

18 |
Partitioning points and graphs to minimize the maximum or the sum of diameters
- Monma, Suri
- 1991
(Show Context)
Citation Context ...on-negative arguments. The 2-clustering problem for the maximum diameter has been treated by Asano, Bhattacharya, Keil, and Yao [1]. They gave an O(n log n) algorithm for this problem. Monma and Suri =-=[16]-=- gave an O(n 2 ) algorithm for finding a 2-clustering with smallest sum of diameters. Some further related results are discussed in the concluding section. Overview of the paper. The key result that w... |

16 | Covering Convex Sets with NonOverlapping Polygons, - Edelsbrunner, Robson - 1990 |

11 | Some packing and covering theorems - T'oth - 1950 |

7 | Tóth, Some packing and covering theorems - Fejes - 1950 |

2 |
Topics in Computational Geometry, Ph
- Supowit
- 1981
(Show Context)
Citation Context ...Is there a k-clustering for S into k sets C1,C2,...,Ck such that F(W (C1),W(C2),...,W(Ck)) ≤ d? Previous work and our result. If k is part of the input, this problem is in general NPcomplete. Supowi=-=t [18]-=- has shown this result for W being the diameter and for F being the maximum function; in other words, for k part of the input, minimizing the maximum diameter in a k-clustering is NP-complete. The rel... |

1 |
Voronoi diagrams in the Laguerre metric and its applications
- Imai, Iri, et al.
- 1985
(Show Context)
Citation Context ...sks. If we partition the plane by assigning every point to the disk Di for which its power d(x, Mi) 2 − r2 i is minimal, we get the so-called power diagram (cf. Aurenhammer [2]; Imai, Iri, and Murot=-=a [12]-=-; or Edelsbrunner [7], section 13.6). It is known that the power diagram is a dissection of the plane into (at most) k convex polygonal regions, very much like in the case of Voronoi diagrams, which a... |