We consider the classic Euclidean k-median and k-means objective on insertion-only streams, where the goal is to maintain a (1+epsilon)-approximation to the k-median or k-means, while using as little memory as possible. Over the last 20 years, clustering in data streams has received a tremendous amount of attention and has been the test-bed for a large variety of new techniques, including coresets, the merge-and-reduce framework, bicriteria approximation, sensitivity sampling, and so on. Despite this intense effort to obtain smaller sketches for these problems, all known techniques require storing at least Omega(log(n Delta)) words of memory, where n is size of the input and Delta is the aspect ratio. A natural question is if one can beat this logarithmic dependence on n and Delta. In this paper, we break this barrier by giving the first algorithm that achieves a (1+epsilon)-approximation to the more general (k,z)-clustering problem, using only ~O(dk/varepsilon^2)*(2^{z log z})*min(1/epsilon^z, k)*poly(log log(n Delta)) words of memory.

Joint work with Vincent Cohen-Addad and David P. Woodruff.


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