[Paper Review] Dynamic k-center clustering with lifetimes

Simone Moretti, Paolo Pellizzoni+2 more|arXiv (Cornell University)|Mar 24, 2026

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TL;DR

The paper introduces dynamic k-center clustering with lifetimes, achieving near 2-approx and 6-approx deterministic guarantees under different update regimes, with sublinear amortized/update-time and memory in active data.

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 $ ilde{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 $ ilde{O}(k/\varepsilon)$ worst-case update time and heavily sublinear working memory.

Motivation & Objective

• Motivate and formalize clustering under a dynamic setting with lifetimes bridging sliding window and fully dynamic models.
• Develop deterministic approximation algorithms for dynamic k-center with lifetimes.
• Analyze update time, memory usage, and compare with prior dynamic clustering approaches in related settings.

Proposed method

• Introduce a dynamic k-center with lifetimes model where each point has an arrival and a known deletion time.
• Maintain multiple radius guesses Γ=(1+β)^i to guide clustering decisions.
• For each γ∈Γ maintain centers C^γ, disjoint point groups D_i^γ, and unclustered U^γ to certify feasibility.
• Use a priority queue Q to track deletions and perform Insert/Delete with structured reclustering to bound work.
• Query returns the centers from the smallest γ with U^γ empty, achieving a (2+ε)-approximation.

Experimental results

Research questions

• RQ1 How to achieve near-optimal (2+ε)-approximation for dynamic k-center with lifetimes with sublinear updates?
• RQ2 Can one design a space-efficient algorithm for the same problem with slightly weaker guarantees?
• RQ3 What are the trade-offs between update time, memory, and approximation when leveraging known lifetimes?
• RQ4 How do lifecycle-aware strategies compare to sliding-window and fully dynamic models in terms of guarantees and resources?

Key findings

• A deterministic (2+ε)-approximation algorithm with Õ(k/ε) amortized update time and Õ(|X|/ε) memory.
• A deterministic (6+ε)-approximation algorithm that, under tame update sequences, has Õ(k/ε) worst-case update time and Õ(k/ε) memory.
• The lifetime information yields improved performance guarantees over state-of-the-art fully dynamic or sliding-window approaches.
• The approach is deterministic and robust against metric-adaptive adversaries compared to some prior results.
• Comparison shows substantial improvements in update time and memory under lifetimes relative to prior methods.