Geometric algorithms for density-based data clustering

We present new geometric approximation and exact algorithms for the density-based data clustering problem in d-dimensional space R-d (for any constant integer d greater than or equal to 2). Previously known algorithms for this problem are efficient only for uniformly-distributed points. However, these algorithms all run in theta(n(2)) time in the worst case, where n is the number of input points. Our approximation algorithm based on the e-fuzzy distance function takes 0(n log n) time for any given fixed value epsilon > 0, and our exact algorithms take sub-quadratic time. The running times and output quality of our algorithms do not depend on any particular data distribution. We believe that our fast approximation algorithm is of considerable practical importance, while our sub-quadratic exact algorithms are more of theoretical interest. We implemented our approximation algorithm and the experimental results show that our approximation algorithm is efficient on arbitrary input point sets.