An Algorithm for Finding Best Matches in Logarithmic Expected Time

Abstract

An algorithm and data structure are presented for searching a file containing N records, each described by k real valued keys, for the m closest matches or nearest neighbors to a given query record.The computation required to organize the file is proportional to kNlogN. The expected number of records examined in each search is independent of the file size. The expected computation to perform each search is proportional-to 1ogN. Empirical evidence suggests that except for very small files, this algorithm is considerably faster than other methods.(Submitted to ACM Transactions on Mathematical Software) Work supported in part by U.S. Energy Research and Development Administration under contract E(O43)515The Best Match or Nearest Neighbor ProblemThe best match or nearest neighbor problem applies to data files that store records with several real valued keys or attributes. The problem is to find those records in the file most similar to a query record according to some dissimilarity or distance measure. Formally, given a file of N recor,ds (each of which is described by k real valued attributes) and a dissimilarity measure D, find the m closest records to a query record (possibly not in the file) with specified attribute values.A data file, for example, might contain information on all cities with post offices. Associated with each city is its longitude and latitude. If a letter is addressed to a town without a post office, the closest town that has a post office might be chosen as the destination.The solution to this problem is of use in many applications. Information retrieval might involve searching a catalog for those items most similar to a given query item; each item in the file would be cataloged by numerical attributes that describe its characteristics. Classification decisions can be made by selecting prototype features from each category and finding which of these prototypes is closest to the record to be classified. Multivariate density estimation can be performed by calculating the volume about a given point ccntaining the closest m neighbors. Structures Used for Associative SearchingOne straightforward technique for solving the best match or nearest neighbor problem is the cell method. The k-dimensional key space is divided into small,identically sized cells. A spiral search of the cells from any query record will find the best matches of that record. Although _ this procedure minimizes the number of records examined, it is extremely costly in space and time, especially when the dimensionality of the space is large.-l- what is termed an optimized k-d tree.-4-The Search AlgorithmThe k-d tree data structure provides an efficient mechanism for examining only those records closest to the query record, thereby greatly reducing the computation required to find the best matches.The search algorithm is most easily described as a recursive procedure. The argument to the procedure is the node under investigation.The first invocation passes the root of the tree as this argument. Available as a global array ...

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