Subquadratic approximation algorithms for clustering problems in high dimensional spaces

Allan Borodin*, Rafail Ostrovsky, Yuval Rabani

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

27 Scopus citations


One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or `nearly quadratic' running time; i.e., O(dn2-α(d)) time where n is the number of data points, d is the dimension of the space and α(d) approaches 0 as d grows. In this paper, we show (for three fairly natural clustering rules) that computing an approximate solution can be done much more efficiently. More specifically, for agglomerative clustering (used, for example, in the Alta VistaTM search engine), for the clustering defined by sparse partitions, and for a clustering based on minimum spanning trees we derive randomized (1+ε) approximation algorithms with running times O(d2n2-γ) where γ>0 depends only on the approximation parameter ε and is independent of the dimension d.

Original languageAmerican English
Pages (from-to)435-444
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
StatePublished - 1999
Externally publishedYes
EventProceedings of the 1999 31st Annual ACM Symposium on Theory of Computing - FCRC '99 - Atlanta, GA, USA
Duration: 1 May 19994 May 1999


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