Accurate low-space approximation of metric k-median for insertion-only streams

We present a low-constant approximation for metric kmedian on an insertion-only stream of n points using O(∈⁻³k log n) space. In particular, we present a streaming (O(∈⁻³k log n),2 + ∈)-bicriterion solution that reports cluster weights. It is well-known that running an offline algorithm on this bicriterion solution yields a (17.66 + ∈)- 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, ∈)-coreset construction requires O(∈⁻² k log⁴ 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 + ∈. We also provide a matching lower bound, showing that any polylog(n)-space streaming algorithm that maintains an (α, β)-bicriterion must have β ≥ 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(∈⁻³k log n),2 + ∈)-bicriterion solution for the entire input.