# Coresets for Minimum Enclosing Balls over Sliding Windows

## TL;DR

This paper introduces efficient algorithms for maintaining coresets for the minimum enclosing ball over sliding windows in data streams, significantly improving speed while preserving approximation quality.

## Contribution

The authors develop the first practical algorithms for sliding-window coresets for MEB, including SWMEB and SWMEB+, with extensions to reproducing kernel Hilbert spaces.

## Key findings

- Achieve up to 10,000x speedup over batch algorithms.
- Maintain small approximation errors for MEB coresets.
- Support for kernels in reproducing kernel Hilbert spaces.

## Abstract

\emph{Coresets} are important tools to generate concise summaries of massive datasets for approximate analysis. A coreset is a small subset of points extracted from the original point set such that certain geometric properties are preserved with provable guarantees. This paper investigates the problem of maintaining a coreset to preserve the minimum enclosing ball (MEB) for a sliding window of points that are continuously updated in a data stream. Although the problem has been extensively studied in batch and append-only streaming settings, no efficient sliding-window solution is available yet. In this work, we first introduce an algorithm, called AOMEB, to build a coreset for MEB in an append-only stream. AOMEB improves the practical performance of the state-of-the-art algorithm while having the same approximation ratio. Furthermore, using AOMEB as a building block, we propose two novel algorithms, namely SWMEB and SWMEB+, to maintain coresets for MEB over the sliding window with constant approximation ratios. The proposed algorithms also support coresets for MEB in a reproducing kernel Hilbert space (RKHS). Finally, extensive experiments on real-world and synthetic datasets demonstrate that SWMEB and SWMEB+ achieve speedups of up to four orders of magnitude over the state-of-the-art batch algorithm while providing coresets for MEB with rather small errors compared to the optimal ones.