Martín Costa

Sayan Bhattacharya, Martín Costa, Ermiya Farokhnejad et al.

We consider the Euclidean $k$-means clustering problem in a dynamic setting, where we have to explicitly maintain a solution (a set of $k$ centers) $S \subseteq \mathbb{R}^d$ subject to point insertions/deletions in $\mathbb{R}^d$. We present a dynamic algorithm for Euclidean $k$-means with $\mathrm{poly}(1/ε)$-approximation ratio, $\tilde{O}(k^ε)$ update time, and $\tilde{O}(1)$ recourse, for any $ε\in (0,1)$, even when $d$ and $k$ are both part of the input. This is the first algorithm to achieve a constant ratio with $o(k)$ update time for this problem, whereas the previous $O(1)$-approximation runs in $\tilde O(k)$ update time [Bhattacharya, Costa, Farokhnejad; STOC'25]. In fact, previous algorithms cannot go beyond $O(k)$ update time precisely because they are designed for general metrics where an $Ω(k)$ lower bound is known. We break this $O(k)$ barrier by devising new fundamental data structures to utilize Euclidean properties: a structure that (implicitly) maintains a clustering subject to both center and data point updates, and a range query structure that can evaluate a mergeable function over any metric ball range given as a query. To obtain these structures, we devise the first consistent hashing scheme [Czumaj, Jiang, Krauthgamer, Vesel{ý}, Yang; FOCS'22] that achieves $\tilde O(n^ε)$ running time per point evaluation with competitive parameters. Our final algorithm exploits the framework of [Bhattacharya, Costa, Farokhnejad; STOC'25] for general metrics. The key change is to redesign several critical subroutines so that they reduce to our new Euclidean data structures, replacing the general-metric implementations that are unlikely to run efficiently even when Euclidean properties are provided.

Martín Costa, Ermiya Farokhnejad

The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total distance from each point in $V$ to its nearest center in $S$. Mettu and Plaxton [UAI'02] gave a randomized algorithm for $k$-median that computes a $O(1)$-approximation in $\tilde O(nk)$ time. They also showed that any algorithm for this problem with a bounded approximation ratio must have a running time of $Ω(nk)$. Thus, the running time of their algorithm is optimal up to polylogarithmic factors. For deterministic $k$-median, Guha et al.~[FOCS'00] gave an algorithm that computes a $\text{poly}(\log (n/k))$-approximation in $\tilde O(nk)$ time, where the degree of the polynomial in the approximation is unspecified. To the best of our knowledge, this remains the state-of-the-art approximation of any deterministic $k$-median algorithm with this running time. This leads us to the following natural question: What is the best approximation of a deterministic $k$-median algorithm with near-optimal running time? We make progress in answering this question by giving a deterministic algorithm that computes a $O(\log(n/k))$-approximation in $\tilde O(nk)$ time. We also provide a lower bound showing that any deterministic algorithm with this running time must have an approximation ratio of $Ω(\log n/(\log k + \log \log n))$, establishing a gap between the randomized and deterministic settings for $k$-median.