On approximating the average distance between points

We consider the problem of approximating the average distance between pairs of points in a high-dimensional Euclidean space, and more generally in any metric space. We consider two algorithmic approaches: 1. Referring only to Euclidean Spaces, we randomly reduce the high-dimensional problem to a one-dimensional problem, which can be solved in time that is almost-linear in the number of points. The resulting algorithm is somewhat better than a related algorithm that can be obtained by using the known randomized embedding of Euclidean Spaces into l1-metric. 2. An alternative approach consists of selecting a random sample of pairs of points and outputting the average distance between these pairs. It turns out that, for any metric space, it suffices to use a sample of size that is linear in the number of points. Our analysis of this method is somewhat simpler and better than the known analysis of Indyk (STOC, 1999). We also study the existence of corresponding deterministic algorithms, presenting both positive and negative results. In particular, in the Euclidean case, this approach outperforms the first approach. In general, the second approach seems superior to the first approach.