Sequential no-Substitution k-Median-Clustering

We study the sample-based $k$-median clustering objective under a sequential setting without substitutions. In this setting, the goal is to select k centers that approximate the optimal clustering on an unknown distribution from a finite sequence of i.i.d. samples, where any selection of a center must be done immediately after the center is observed and centers cannot be substituted after selection. We provide an efficient algorithm for this setting, and show that its multiplicative approximation factor is twice the approximation factor of an efficient offline algorithm. In addition, we show that if efficiency requirements are removed, there is an algorithm that can obtain the same approximation factor as the best offline algorithm.