A Constant Approximation Algorithm for Sequential Random-Order No-Substitution k-Median Clustering
This work provides a significant theoretical advancement in online clustering algorithms for data stream processing, particularly for applications where immediate and irreversible decisions on cluster centers are required.
This paper addresses the k-median clustering problem in a sequential, no-substitution setting where data points arrive as a stream and centers must be chosen immediately and cannot be changed. The authors developed the first algorithm that achieves a constant approximation factor for the optimal risk under a random arrival order, which is an exponential improvement over prior work.
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.