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

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 a new algorithm for this setting that obtains a constant approximation factor on the optimal risk under a random arrival order. This is the first such algorithm that holds without any assumptions on the input data and selects a non-trivial number of centers. The number of selected centers is 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 repeated calculations using an offline clustering algorithm.

Fast Distributed k-Means with a Small Number of Rounds

“…When all the points are removed, the algorithm stops and calculates the final clustering from the centers selected in the centralized clustering runs. SOCCER combines clustering approaches designed for two different settings: The distributed setting (Ene et al, 2011) and the online setting (Hess et al, 2021). Ene et al (2011) iteratively samples points from the machines and then removes points that are close to them from consideration.…”

“…Ene et al (2011) iteratively samples points from the machines and then removes points that are close to them from consideration. We show that calculating a clustering on the point sample, along with a technique adapted from Hess et al (2021), lead to a more accurate removal of points. This provides a practical and successful algorithm with approximation guarantees that depend only on the number of points that the coordinator can cluster.…”

“…This estimate is then used to calculate the threshold that the machines use to remove points from their dataset. The method for estimating the cost and calculating the threshold is based on a technique first proposed in Hess et al (2021), which addresses a different setting of (centralized) online no-substitution clustering. In that work, the estimate is used for the purpose of on-the-fly center selection, when clustering a stream of points.…”

“…The improvement of the constants is of significant practical importance: These constants are used by SOCCER. If they were too large, as in Hess et al (2021), then SOCCER would be impractical. For instance, in Hess et al (2021), the number of outliers removed when calculating the truncated cost is very large.…”

“…If they were too large, as in Hess et al (2021), then SOCCER would be impractical. For instance, in Hess et al (2021), the number of outliers removed when calculating the truncated cost is very large. Using the same number in SOCCER would have caused the fraction of removed points in each round to be too small, leading to a large number of rounds.…”