Dynamic Algorithm for Explainable $k$-medians Clustering under ℓp Norm

We study the problem of explainable k-medians clustering introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (2020). In this problem, the goal is to construct a threshold decision tree that partitions data into k clusters while minimizing the k-medians objective. These trees are interpretable because each internal node makes a simple decision by thresholding a single feature, allowing users to trace and understand how each point is assigned to a cluster. We present the first algorithm for explainable k-medians under lp norm for every finite p>= 1. Our algorithm achieves an O(p(log k)^1 + 1/p - 1/p^2) approximation to the optimal k-medians cost for any p>= 1. Previously, algorithms were known only for p = 1 and p = 2. For p = 2, our algorithm improves upon the existing bound of O(log^3/2k), and for p = 1, it matches the tight bound of log k + O(1) up to a multiplicative O(log log k) factor. We show how to implement our algorithm in a dynamic setting. The dynamic algorithm maintains an explainable clustering under a sequence of insertions and deletions, with amortized update time O(d log^3 k) and O(log k) recourse, making it suitable for large-scale and evolving datasets.