A Unified Approach for Clustering Problems on Sliding Windows

We explore clustering problems in the streaming sliding window model in both general metric spaces and Euclidean space. We present the first polylogarithmic space $O(1)$-approximation to the metric $k$-median and metric $k$-means problems in the sliding window model, answering the main open problem posed by Babcock, Datar, Motwani and OCallaghan, which has remained unanswered for over a decade. Our algorithm uses $O(k^3 log^6 n)$ space and $operatorname{poly}(k, log n)$ update time. This is an exponential improvement on the space required by the technique due to Babcock, et al. We introduce a data structure that extends smooth histograms as introduced by Braverman and Ostrovsky to operate on a broader class of functions. In particular, we show that using only polylogarithmic space we can maintain a summary of the current window from which we can construct an $O(1)$-approximate clustering solution. Merge-and-reduce is a generic method in computational geometry for adapting offline algorithms to the insertion-only streaming model. Several well-known coreset constructions are maintainable in the insertion-only streaming model using this method, including well-known coreset techniques for the $k$-median, $k$-means in both low-and high-dimensional Euclidean spaces. Previous work has adapted these techniques to the insertion-deletion model, but translating them to the sliding window model has remained a challenge. We give the first algorithm that, given an insertion-only streaming coreset construction of space $s$, maintains a $(1pmepsilon)$-approximate coreset in the sliding window model using $O(s^2epsilon^{-2}log n)$ space. For clustering problems, our results constitute the first significant step towards resolving problem number 20 from the List of Open Problems in Sublinear Algorithms.

AMS Without 4-Wise Independence on Product Domains

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 pm epsilon)$ approximation (with probability $1-delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.