Streaming Balanced Clustering

TL;DR

This paper introduces the first single-pass streaming algorithm for capacitated clustering problems, including k-median and k-means, that handles insertions and deletions with near-capacity constraints in high-dimensional Euclidean space.

Contribution

It presents a novel single-pass streaming algorithm for balanced clustering that handles both insertions and deletions, using space poly$(k d ext{log} \Delta)$ and a new space decomposition technique.

Findings

Developed a space decomposition via curved half-spaces.

Designed a strong coreset of size poly$(k d ext{log} \Delta)$.

Algorithm handles insertions and deletions with capacity violation only by a $1+ ext{epsilon}$ factor.

Abstract

Clustering of data points in metric space is among the most fundamental problems in computer science with plenty of applications in data mining, information retrieval and machine learning. Due to the necessity of clustering of large datasets, several streaming algorithms have been developed for different variants of clustering problems such as $k$-median and $k$-means problems. However, despite the importance of the context, the current understanding of balanced clustering (or more generally capacitated clustering) in the streaming setting is very limited. The only previously known streaming approximation algorithm for capacitated clustering requires three passes and only handles insertions. In this work, we develop \emph{the first single pass streaming algorithm} for a general class of clustering problems that includes capacitated $k$-median and capacitated $k$-means in Euclidean space, using only poly$( k d \log \Delta)$ space, where $k$ is the number of clusters, $d$ is the dimension and $\Delta$ is the maximum relative range of a coordinate. (Note that $d\log \Delta$ is the space required to represent one point.) This algorithm only violates the capacity constraint by a $1+\epsilon$ factor. Interestingly, unlike the previous algorithm, our algorithm handles both insertions and deletions of points. To provide this result we define a decomposition of the space via some curved half-spaces. We used this decomposition to design a strong coreset of size poly$( k d \log \Delta)$ for balanced clustering. Then, we show that this coreset is implementable in the streaming and distributed settings.