Researchers have created a fully dynamic algorithm for the metric k‑median problem that maintains a constant‑factor approximation while guaranteeing provable limits on how long each update takes and how many centers can shift. Each time a client is added or removed, the algorithm needs only about k time, and at most (1+log Δ)³ centers can be reassigned, where Δ is the aspect ratio of the underlying metric. The algorithm also works for k‑means under the same asymptotic bounds.

The method achieves these results by carefully sampling and updating a small set of candidate centers. After an initial setup that takes ≈ k (n₀+k) time for the starting clients, every subsequent insertion or deletion requires only Õ(k) operations. The worst‑case analysis shows that even the largest possible change in the set of centers is bounded by O((1+log Δ)³). These bounds hold for every execution, independent of random choices, while the constant‑factor approximation guarantee holds with high probability over any fixed sequence of updates.

Compared with earlier schemes, the new approach eliminates the previous dependence on amortized analysis. Earlier algorithms offered only amortized update times and logarithmic recourse, whereas this work provides worst‑case guarantees for both.