Optimal algorithms for approximate clustering

@inproceedings{Feder1988OptimalAF,
  title={Optimal algorithms for approximate clustering},
  author={Tom{\'a}s Feder and Daniel H. Greene},
  booktitle={STOC '88},
  year={1988}
}
In a clustering problem, the aim is to partition a given set of <italic>n</italic> points in <italic>d</italic>-dimensional space into <italic>k</italic> groups, called clusters, so that points within each cluster are near each other. Two objective functions frequently used to measure the performance of a clustering algorithm are, for any <italic>L<subscrpt>4</subscrpt></italic> metric, (a) the maximum distance between pairs of points in the same cluster, and (b) the maximum distance between… Expand
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References

SHOWING 1-10 OF 34 REFERENCES
An optimal algorithm for the all-nearest-neighbors problem
  • P. M. Vaidya
  • Mathematics, Computer Science
  • 27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
  • 1986
TLDR
This work gives an O(nlogn) algorithm for the All-Nearest-Neighbors problem, for fixed dimension k and fixed metric Lq, and shows that the running time of this algorithm is optimal upto a constant. Expand
Clustering to Minimize the Maximum Intercluster Distance
  • T. Gonzalez
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1985
TLDR
An O(kn) approximation algorithm that guarantees solutions with an objective function value within two times the optimal solution value is presented and it is shown that this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality. Expand
A Best Possible Heuristic for the k-Center Problem
TLDR
A 2-approximation algorithm for the k-center problem with triangle inequality is presented, the key combinatorial object used is called a strong stable set, and the NP-completeness of the corresponding decision problem is proved. Expand
Optimal Packing and Covering in the Plane are NP-Complete
TLDR
This paper proves that even severely restricted instances of packing and covering problems remain NP-hard in two or more dimensions, and helps to fill the gap by showing that some very constrained intersection graph problems in two dimensions are not very constrained. Expand
Approximation Algorithms for NP-Complete Problems on Planar Graphs (Preliminary Version)
  • B. Baker
  • Mathematics, Computer Science
  • FOCS
  • 1983
TLDR
A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand
On the Complexity of Clustering Problems
The class of discrete optimization problems may be partitioned into two subclasses PS and P?. PS is the subclass of problems which are known to be polynomial solvable. P? is the subclass of problemsExpand
On the Complexity of Some Common Geometric Location Problems
TLDR
The p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard and the reductions are from 3-satisfiability. Expand
Easy and hard bottleneck location problems
TLDR
This work considers a bottleneck location problem on a graph and presents an efficient (polynomial time) algorithm for solving it, and shows that two other bottleneck location problems, finding K -centers and absolute K - centres of a graph appear to be very difficult to solve even for reasonably good approximate solutions. Expand
A unified approach to approximation algorithms for bottleneck problems
In this paper a powerful, and yet simple, technique for devising approximation algorithms for a wide variety of NP-complete problems in routing, location, and communication network design isExpand
Fast algorithms for vector quantization picture coding
  • W. Equitz
  • Computer Science, Mathematics
  • ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing
  • 1987
TLDR
A data structure (k-d trees, developed by Bentley) is demonstrated to be appropriate for implementing exact nearest neighbor searching in time logarithmic in codebook size and is generalizable to any vector quantization application with the appropriate distortion measure. Expand
...
1
2
3
4
...