Accurate Low-Space Approximation of Metric k-Median for Insertion-Only Streams

Abstract

We present a low-constant approximation for metric k-median on an insertion-only stream of n points using $O(\epsilon ^{-3} k \log n)$ space. In particular, we present a streaming $(O(\epsilon ^{-3} k \log n), 2 + \epsilon )$-bicriterion solution that reports cluster weights. It is well-known that running an offline algorithm on this bicriterion solution yields a $(17.66 + \epsilon )$-approximation.

Previously, there have been two lines of research that trade off between space and accuracy in the streaming k-median problem. To date, the best-known $(k,\epsilon )$-coreset construction requires $O(\epsilon ^{-2} k \log ^4 n)$ space [8], while the best-known $O(k \log n)$-space algorithm provides only a $(O(k \log n), 1063)$-bicriterion [3]. Our work narrows this gap significantly, matching the best-known space while significantly improving the accuracy from 1063 to $2+\epsilon$. We also provide a matching lower bound, showing that any ${\text {polylog}}(n)$-space streaming algorithm that maintains an $(\alpha ,\beta )$-bicriterion must have $\beta \ge 2$.

Our technique breaks the stream into segments defined by jumps in the optimal clustering cost, which increases monotonically as the stream progresses. By a storing an accurate summary of recent segments and a lower-space summary of older segments, our algorithm maintains a $(O(\epsilon ^{-3} k \log n), 2 + \epsilon )$-bicriterion solution for the entire input.

V. Braverman—This material is based upon work supported in part by the National Science Foundation under Grants IIS-1447639 and CCF-1650041.

H. Lang—This research is supported by the Franco-American Fulbright Commission. The author thanks INRIA (l’Institut national de recherche en informatique et en automatique) for hosting him during the writing of this paper.