Sublinear time algorithms for metric space problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating average parameters of graphs
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Expanders with respect to Hadamard spaces and random graphs: extended abstract
Proceedings of the 5th conference on Innovations in theoretical computer science
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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: 1Referring 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 茂戮驴1-metric.1An 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.