Approximate Nearest Neighbor Algorithms for Hausdorff Metrics via Embeddings

Hausdorff metrics are used in geometric settings for measuring the distance between sets of points. They have been used extensively in areas such as computer vision, pattern recognition and computational chemistry. While computing the distance between a single pair of sets under the Hausdorff metric has been well studied, no results were known for the Nearest Neighbor problem under Hausdorff metrics. Indeed, no results were known for the nearest neighbor problem for any metric without norm structure, of which the Hausdorff is one. We present the first nearest neighbor algorithm for the Hausdorff metric. We achieve our result by embedding Hausdorff metrics into l∞ and using known nearest neighbor algorithms for this target metric. We give upper and lower bounds on the number of dimensions needed for such an l∞ embedding. Our bounds require the introduction of new techniques based on superimposed codes and non-uniform sampling. ∗Department of Computer Science, Rutgers University, Piscataway, NJ 08855, USA. ([email protected], http://www.cs.rutgers.edu/∼farach). Supported by NSF Career Development Award CCR-9501942, NATO Grant CRG 960215 and an Alfred P. Sloan Research Fellowship. †Supported by Stanford Graduate Fellowship and ARO MURI Grant DAAH04–96–1–0007. Part of this work was done while the author was visiting Bell Laboratories.