A Constant Approximation Algorithm for Sequential Random-Order No-Substitution k-Median Clustering

TL;DR

This paper introduces a novel constant approximation algorithm for sequential no-substitution k-median clustering in random data streams, achieving significant improvements over prior work without structural data assumptions.

Contribution

It presents the first constant approximation algorithm for this setting with random order, using new risk estimation and multiscale center selection techniques.

Findings

Achieves constant approximation factor under random arrival order.

Number of centers is quasi-linear in k.

First guarantee without structural assumptions on data.

Abstract

We study k-median clustering under the sequential no-substitution setting. In this setting, a data stream is sequentially observed, and some of the points are selected by the algorithm as cluster centers. However, a point can be selected as a center only immediately after it is observed, before observing the next point. In addition, a selected center cannot be substituted later. We give the first algorithm for this setting that obtains a constant approximation factor on the optimal risk under a random arrival order, an exponential improvement over previous work. This is also the first constant approximation guarantee that holds without any structural assumptions on the input data. Moreover, the number of selected centers is only quasi-linear in k. Our algorithm and analysis are based on a careful risk estimation that avoids outliers, a new concept of a linear bin division, and a multiscale approach to center selection.