Dynamic k-center clustering with lifetimes

TL;DR

This paper introduces a new dynamic clustering setting with known point lifetimes, and provides efficient deterministic algorithms with strong approximation guarantees for this setting.

Contribution

It proposes the first algorithms for clustering with lifetimes, bridging sliding window and fully dynamic models, with improved update times and memory efficiency.

Findings

Deterministic (2+ε)-approximation algorithm with near-linear amortized update time.

Deterministic (6+ε)-approximation algorithm with sublinear memory under tame sequences.

Algorithms effectively handle dynamic data with known lifetimes, improving efficiency over previous models.

Abstract

The $k$-center problem is a fundamental clustering variant with applications in learning systems and data summarization. In several real-world scenarios, the dataset to be clustered is not static, but evolves over time, as new data points arrive and old ones become stale. To account for dynamicity, the $k$-center problem has been mainly studied under the sliding window setting, where only the $N$ most recent points are considered non-stale, or the fully dynamic setting, where arbitrary sequences of point arrivals and deletions without prior notice may occur. In this paper, we introduce the dynamic setting with lifetimes, which bridges the two aforementioned classical settings by still allowing arbitrary arrivals and deletions, but making the deletion time of each point known upon its arrival. Under this new setting, we devise a deterministic $(2+\varepsilon)$-approximation algorithm with $\tilde{O}(k/\varepsilon)$ amortized update time and memory usage linear in the number of currently active points. Moreover, we develop a deterministic $(6+\varepsilon)$-approximation algorithm that, under tame update sequences, has $\tilde{O}(k/\varepsilon)$ worst-case update time and heavily sublinear working memory.