Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Approximate closest-point queries in high dimensions
Information Processing Letters
A Delaunay refinement algorithm for quality 2-dimensional mesh generation
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
Fast hierarchical clustering and other applications of dynamic closest pairs
Journal of Experimental Algorithmics (JEA)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Parallel Construction of Quadtrees and Quality Triangulations
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Performance Guarantees for Hierarchical Clustering
COLT '02 Proceedings of the 15th Annual Conference on Computational Learning Theory
On the completeness of a generalized matching problem
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
The skip quadtree: a simple dynamic data structure for multidimensional data
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Tightening non-simple paths and cycles on surfaces
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Algorithms on negatively curved spaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
Packing and Covering δ-Hyperbolic Spaces by Balls
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
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We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.