Relational Algorithms for k-means Clustering

Benjamin Moseley, Kirk Pruhs, Alireza Samadian, Yuyan Wang

Abstract:

This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an
algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows
represent the data points. The running time is potentially exponentially smaller than N, the number of data points to be
clustered that the relational database represents.

Few relational algorithms are known and this paper offers techniques for designing relational algorithms as well as
characterizing their limitations. We show that given two data points as cluster centers, if we cluster points according to
their closest centers, it is NP-Hard to approximate the number of points in the clusters on a general relational input.
This is trivial for conventional data inputs and this result exemplifies that standard algorithmic techniques may not
be directly applied when designing an efficient relational algorithm. This paper then introduces a new method that
leverages rejection sampling and the k-means++ algorithm to construct a O(1)-approximate k-means solution.