Chromatic Nearest Neighbor Searching A Query Sensitive Approach.pdf

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Chromatic Nearest Neighbor Searching A Query Sensitive Approach.pdf

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Chromatic Nearest Neighbor Searching A Query Sensitive Approach

Chromatic Nearest Neighbor Searching: A Query Sensitive Approach? David M. Mount? Nathan S. Netanyahu? Ruth Silverman§ Angela Y. Wu? Last modified: October 21, 1996. (Minor updates made: August 24, 2000.) Final version for CGTA. Abstract The nearest neighbor problem is that of preprocessing a set P of n data points in Rd so that, given any query point q, the closest point in P to q can be determined efficiently. In the chromatic nearest neighbor problem, each point of P is assigned a color, and the problem is to determine the color of the near- est point to the query point. More generally, given k ≥ 1, the problem is to determine the color occurring most frequently among the k nearest neighbors. The chromatic version of the nearest neighbor problem is used in many appli- cations in pattern recognition and learning. In this paper we present a simple algorithm for solving the chromatic k nearest neighbor problem. We provide a query sensitive analysis, which shows that if the color classes form spatially well separated clusters (as often happens in practice), then queries can be an- swered quite efficiently. We also allow the user to specify an error bound ≥ 0, ?A preliminary version of this paper appeared in the Proceedings of the 7th Canadian Conference on Computational Geometry, 1995, 261–266. ?Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Email: mount@cs.umd.edu. The support of the National Science Foundation under grant CCR–9310705 is gratefully acknowledged. ?Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel, and Center for Automation Research, University of Maryland, College Park, Maryland 20742. This research was carried out while the author was also affiliated with the Center of Excellence in Space Data and Information Sciences at NASA Goddard Space Flight Center, Greenbelt, MD 20771. Email: nathan@macs.biu.ac.il. §Center for Automation Re