A @math -nearest neighbor ( @math NN) query determines the @math nearest points, using distance metrics, from a specific location. An all @math -nearest neighbor (A @math NN) query constitutes a variation of a @math NN query and retrieves the @math nearest points for each point inside a database. Their main usage resonates in spatial databases and they consist the backbone of many location-based applications and not only (i.e. @math NN joins in databases, classification in data mining). So, it is very crucial to develop methods that answer them efficiently. In this work, we propose a novel method for classifying multidimensional data using an A @math NN algorithm in the MapReduce framework. Our approach exploits space decomposition techniques for processing the classification procedure in a parallel and distributed manner. To our knowledge, we are the first to study the classification of multidimensional objects under this perspective. Through an extensive experimental evaluation we prove that our solution is efficient and scalable in processing the given queries. We investigate many different perspectives that can affect the total computational cost, such as different dataset distributions, number of dimensions, growth of @math value and granularity of space decomposition and prove that our system is efficient, robust and scalable.
In @cite_9 a minimum spanning tree based classification model is introduced and it can be viewed as an intermediate model between the traditional @math -nearest neighbor method and cluster based classification method. Another approach presented in @cite_2 recommends parallel implementation methods of several classification algorithms, including @math -nearest neighbor, bayesian model, decision tree, but does not contemplate the nor the perspective of dimensionality nor parameter @math .
@cite_9 @cite_2
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Rapid growth of data has provided us with more information, yet challenges the tradition techniques to extract the useful knowledge. In this paper, we propose MCMM, a Minimum spanning tree (MST) based Classification model for Massive data with MapReduce implementation. It can be viewed as an intermediate model between the traditional K nearest neighbor method and cluster based classification method, aiming to overcome their disadvantages and cope with large amount of data. Our model is implemented on Hadoop platform, using its MapReduce programming framework, which is particular suitable for cloud computing. We have done experiments on several data sets including real world data from UCI repository and synthetic data, using Downing 4000 clusters, installed with Hadoop. The results show that our model outperforms KNN and some other classification methods on a general basis with respect to accuracy and scalability. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. For some positive constant , we give a (3 2- )-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3 2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.