Can anyone throw any light on any PTAS algorithm that I can apply for K-Clustering algorithm when the distance computation between the clustering points is costly.
In details, I have a set of N points and I have a customized distance metric between them. I want to compute a set of k clusters such that the cluster centers are far apart while among the clusters the points are very close. This customized distance metric is costly to calculate and hence I would like them to be calculated as less as it can possibly be.
I can give more clarification if required.
[EDIT] So the distance metric I have defined is $D(x,y) = 1 - \rho(x,y)$ where $\rho(x,y)$ is the Pearson Correlation Coefficient between points $x$ and $y$ ( $x$ and $y$ are data sets in themselves). Obviously to compute all such Pearson Correlations Coefficient when the number of data-points is 1000 (each set $x$ has another 1000 or so values within themselves) or so can be very expensive computationally.