M-tree

Abstract: A new access method, called M-tree, is proposed to organize and search large data sets from a generic “metric space”, i.e. where object proximity is only defined by a distance function satisfying the positivity, symmetry, and triangle inequality postulates. We detail algorithms for insertion of objects and split management, which keep the M-tree always balanced - several heuristic split alternatives are considered and experimentally evaluated. Algorithms for similarity (range and k-nearest neighbors) queries are also described. Results from extensive experimentation with a prototype system are reported, considering as the performance criteria the number of page I/O’s and the number of distance computations. The results demonstrate that the Mtree indeed extends the domain of applicability beyond the traditional vector spaces, performs reasonably well in high-dimensional data spaces, and scales well in case of growing files.

Abstract: The problem of searching the elements of a set that are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather general case where the similarity criterion defines a metric space, instead of the more restricted case of a vector space. Many solutions have been proposed in different areas, in many cases without cross-knowledge. Because of this, the same ideas have been reconceived several times, and very different presentations have been given for the same approaches. We present some basic results that explain the intrinsic difficulty of the search problem. This includes a quantitative definition of the elusive concept of "intrinsic dimensionality." We also present a unified view of all the known proposals to organize metric spaces, so as to be able to understand them under a common framework. Most approaches turn out to be variations on a few different concepts. We organize those works in a taxonomy that allows us to devise new algorithms from combinations of concepts not noticed before because of the lack of communication between different communities. We present experiments validating our results and comparing the existing approaches. We finish with recommendations for practitioners and open questions for future development.

Abstract: The k-means algorithm is by far the most widely used method for discovering clusters in data. We show how to accelerate it dramatically, while still always computing exactly the same result as the standard algorithm. The accelerated algorithm avoids unnecessary distance calculations by applying the triangle inequality in two different ways, and by keeping track of lower and upper bounds for distances between points and centers. Experiments show that the new algorithm is effective for datasets with up to 1000 dimensions, and becomes more and more effective as the number k of clusters increases. For k ≥ 20 it is many times faster than the best previously known accelerated k-means method.

Abstract: Existing studies on time series are based on two categories of distance functions. The first category consists of the Lp-norms. They are metric distance functions but cannot support local time shifting. The second category consists of distance functions which are capable of handling local time shifting but are nonmetric. The first contribution of this paper is the proposal of a new distance function, which we call ERP ("Edit distance with Real Penalty"). Representing a marriage of L1- norm and the edit distance, ERP can support local time shifting, and is a metric. The second contribution of the paper is the development of pruning strategies for large time series databases. Given that ERP is a metric, one way to prune is to apply the triangle inequality. Another way to prune is to develop a lower bound on the ERP distance. We propose such a lower bound, which has the nice computational property that it can be efficiently indexed with a standard B+- tree. Moreover, we show that these two ways of pruning can be used simultaneously for ERP distances. Specifically, the false positives obtained from the B+-tree can be further minimized by applying the triangle inequality. Based on extensive experimentation with existing benchmarks and techniques, we show that this combination delivers superb pruning power and search time performance, and dominates all existing strategies.