Dynamic Diameter in High-Dimensions against Adaptive Adversary and Beyond

TL;DR

This work tackles dynamic maintenance of diameter and $k$-center clustering for a point set in high-dimensional space under insertions and deletions, under a strong randomness-adaptive adversary. It introduces an $ ext{$\varepsilon$-robust representative}$ built from centerpoints to achieve a guaranteed $2$-approximation for the dynamic diameter with worst-case update time $O\left(d^5 \log^{1.5} d \log^{1.5}\left(\frac{n}{\delta}\right)\right)$, and extends the approach to the minimum enclosing ball. For $k$-center, it gives a $(4+\varepsilon)$-approximation against the same adversary with amortized update time $O\left(k^{2.5} d \cdot {\rm poly}(\log n, \varepsilon^{-1}, \delta)\right)$, via robust centers, sampling, and a de-amortization framework. Collectively, these results deliver the first efficient fully dynamic diameter algorithm in high dimensions that is robust to adaptive adversaries, and they advance robust dynamic clustering in high-dimensional settings with provable guarantees.

Abstract

In this paper, we study the fundamental problems of maintaining the diameter and a $k$-center clustering of a dynamic point set $P \subset \mathbb{R}^d$, where points may be inserted or deleted over time and the ambient dimension $d$ is not constant and may be high. Our focus is on designing algorithms that remain effective even in the presence of an adaptive adversary -- an adversary that, at any time $t$, knows the entire history of the algorithm's outputs as well as all the random bits used by the algorithm up to that point. We present a fully dynamic algorithm that maintains a $2$-approximate diameter with a worst-case update time of $\text{poly}(d, \log n)$, where $n$ is the length of the stream. Our result is achieved by identifying a robust representative of the dataset that requires infrequent updates, combined with a careful deamortization. To the best of our knowledge, this is the first efficient fully-dynamic algorithm for diameter in high dimensions that simultaneously achieves a 2-approximation guarantee and robustness against an adaptive adversary. We also give an improved dynamic $(4+ε)$-approximation algorithm for the $k$-center problem, also resilient to an adaptive adversary. Our clustering algorithm achieves an amortized update time of $k^{2.5} d \cdot \text{poly}(ε^{-1}, \log n)$, improving upon the amortized update time of $k^6 d \cdot \text{poly}(ε^{-1}, \log n)$ by Biabani et al. [NeurIPS'24].

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: Diameter Problem
• Definition 1.4: $k$-Center Problem
• Definition 1.5: Dynamic $\alpha$-Approximate $k$-Center against an Adaptive Adversary
• Definition 2.1: Representative Point
• Definition 2.2: $\epsilon$-Robust Representative Point
• Lemma 2.3
• Definition 2.4: Tukey Depth T75
• Definition 2.5: $\alpha$-centerpoint