A Unified Approach for Clustering Problems on Sliding Windows

TL;DR

This paper introduces a novel, space-efficient streaming algorithm for clustering in sliding window models, achieving the first polylogarithmic space $O(1)$-approximation for metric $k$-median and $k$-means, solving a decade-old open problem.

Contribution

It presents the first polylogarithmic space $O(1)$-approximation algorithms for clustering in sliding window models, extending smooth histograms and coreset techniques to this setting.

Findings

Achieved polylogarithmic space $O(1)$-approximation for metric $k$-median and $k$-means.

Developed a data structure extending smooth histograms for broader functions.

Maintained approximate coresets in sliding windows with $O(s^2 ext{epsilon}^{-2} ext{log} n)$ space.

Abstract

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 O'Callaghan, 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 $(1\pm\epsilon)$-approximate coreset in the sliding window model using $O(s^2\epsilon^{-2}\log n)$ 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.