An incremental Hausdorff distance calculation algorithm

The Hausdorff distance is commonly used as a similarity measure between two point sets. Using this measure, a set X is considered similar to Y iff every point in X is close to at least one point in Y. Formally, the Hausdorff distance HausDist(X, Y) can be computed as the Max-Min distance from X to Y, i.e., find the maximum of the distance from an element in X to its nearest neighbor (NN) in Y. Although this is similar to the closest pair and farthest pair problems, computing the Hausdorff distance is a more challenging problem since its Max-Min nature involves both maximization and minimization rather than just one or the other. A traditional approach to computing HausDist(X, Y) performs a linear scan over X and utilizes an index to help compute the NN in Y for each x in X. We present a pair of basic solutions that avoid scanning X by applying the concept of aggregate NN search to searching for the element in X that yields the Hausdorff distance. In addition, we propose a novel method which incrementally explores the indexes of the two sets X and Y simultaneously. As an example application of our techniques, we use the Hausdorff distance as a measure of similarity between two trajectories (represented as point sets). We also use this example application to compare the performance of our proposed method with the traditional approach and the basic solutions. Experimental results show that our proposed method outperforms all competitors by one order of magnitude in terms of the tree traversal cost and total response time.