We propose an efficient linear-time graph-based divisive cluster analysis approach called Sampling Clustering. It constructs a lite informative dendrogram by recursively dividing a graph into subgraphs. In each recursive call, a graph is sampled first with a set of vertices being removed to disconnect latent clusters, then condensed by adding edges to the remaining vertices to avoid graph fragmentation caused by vertex removals. We also present some sampling and condensing methods and discuss the effectiveness in this paper. Our implementations run in linear time and achieve outstanding performance on various types of datasets. Experimental results show that they outperform state-of-the-art clustering algorithms with significantly less computing resource requirements.
The main purpose of this paper is to describe a process for partitioning an N-dimensional population into k sets on the basis of a sample. The process, which is called 'k-means,' appears to give partitions which are reasonably efficient in the sense of within-class variance. That is, if p is the probability mass function for the population, S = S1, S2, * *, Sk is a partition of EN, and ui, i = 1, 2, * , k, is the conditional mean of p over the set Si, then W2(S) = ff=ISi f z u42 dp(z) tends to be low for the partitions S generated by the method. We say 'tends to be low,' primarily because of intuitive considerations, corroborated to some extent by mathematical analysis and practical computational experience. Also, the k-means procedure is easily programmed and is computationally economical, so that it is feasible to process very large samples on a digital computer. Possible applications include methods for similarity grouping, nonlinear prediction, approximating multivariate distributions, and nonparametric tests for independence among several variables. In addition to suggesting practical classification methods, the study of k-means has proved to be theoretically interesting. The k-means concept represents a generalization of the ordinary sample mean, and one is naturally led to study the pertinent asymptotic behavior, the object being to establish some sort of law of large numbers for the k-means. This problem is sufficiently interesting, in fact, for us to devote a good portion of this paper to it. The k-means are defined in section 2.1, and the main results which have been obtained on the asymptotic behavior are given there. The rest of section 2 is devoted to the proofs of these results. Section 3 describes several specific possible applications, and reports some preliminary results from computer experiments conducted to explore the possibilities inherent in the k-means idea. The extension to general metric spaces is indicated briefly in section 4. The original point of departure for the work described here was a series of problems in optimal classification (MacQueen [9]) which represented special The k-means method is a widely used clustering technique that seeks to minimize the average squared distance between points in the same cluster. Although it offers no accuracy guarantees, its simplicity and speed are very appealing in practice. By augmenting k-means with a very simple, randomized seeding technique, we obtain an algorithm that is Θ(logk)-competitive with the optimal clustering. Preliminary experiments show that our augmentation improves both the speed and the accuracy of k-means, often quite dramatically. We propose a theoretically and practically improved density-based, hierarchical clustering method, providing a clustering hierarchy from which a simplified tree of significant clusters can be constructed. For obtaining a “flat” partition consisting of only the most significant clusters (possibly corresponding to different density thresholds), we propose a novel cluster stability measure, formalize the problem of maximizing the overall stability of selected clusters, and formulate an algorithm that computes an optimal solution to this problem. We demonstrate that our approach outperforms the current, state-of-the-art, density-based clustering methods on a wide variety of real world data.  Clustering algorithms are attractive for the task of class identification in spatial databases. However, the application to large spatial databases rises the following requirements for clustering algorithms: minimal requirements of domain knowledge to determine the input parameters, discovery of clusters with arbitrary shape and good efficiency on large databases. The well-known clustering algorithms offer no solution to the combination of these requirements. In this paper, we present the new clustering algorithm DBSCAN relying on a density-based notion of clusters which is designed to discover clusters of arbitrary shape. DBSCAN requires only one input parameter and supports the user in determining an appropriate value for it. We performed an experimental evaluation of the effectiveness and efficiency of DBSCAN using synthetic data and real data of the SEQUOIA 2000 benchmark. The results of our experiments demonstrate that (1) DBSCAN is significantly more effective in discovering clusters of arbitrary shape than the well-known algorithm CLARANS, and that (2) DBSCAN outperforms CLARANS by a factor of more than 100 in terms of efficiency.  Finding useful patterns in large datasets has attracted considerable interest recently, and one of the most widely studied problems in this area is the identification of clusters, or densely populated regions, in a multi-dimensional dataset. Prior work does not adequately address the problem of large datasets and minimization of I O costs.This paper presents a data clustering method named BIRCH (Balanced Iterative Reducing and Clustering using Hierarchies), and demonstrates that it is especially suitable for very large databases. BIRCH incrementally and dynamically clusters incoming multi-dimensional metric data points to try to produce the best quality clustering with the available resources (i.e., available memory and time constraints). BIRCH can typically find a good clustering with a single scan of the data, and improve the quality further with a few additional scans. BIRCH is also the first clustering algorithm proposed in the database area to handle "noise" (data points that are not part of the underlying pattern) effectively.We evaluate BIRCH's time space efficiency, data input order sensitivity, and clustering quality through several experiments. We also present a performance comparisons of BIRCH versus CLARANS, a clustering method proposed recently for large datasets, and show that BIRCH is consistently superior.