Dynamic Diameter in High-Dimensions against Adaptive Adversary and Beyond

2025-11-02

AI Summary

This paper addresses the robust maintenance of diameter and $k$-center clustering for high-dimensional dynamic point sets under insertions and deletions, adversarially adaptive to the full algorithmic history and internal randomness. We propose the first worst-case poly$(d, log n)$-time algorithm achieving a 2-approximation for diameter, establishing the first theoretical robustness guarantee against adaptive adversaries in high-dimensional dynamic settings. Concurrently, we design a $(4+varepsilon)$-approximate $k$-center algorithm with amortized update time $k^{2.5} d cdot ext{poly}(varepsilon^{-1}, log n)$, substantially improving upon prior $k^6$-dependence. Our technical contributions include: (i) robust representative point selection ensuring stability under adaptive attacks; (ii) a novel de-amortization framework; and (iii) an adaptive-resilient randomization mechanism. Together, these techniques reconcile high-dimensional geometric stability with efficient dynamic updates, advancing the state of robust geometric data structures.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Intelligent Robots: Other Foundations and Applications

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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 $ ext{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 $k^{2.5} d cdot ext{poly}(ε^{-1}, log n)$, improving upon the amortized update time of $k^6 d cdot ext{poly}(ε^{-1}, log n)$ by Biabani et al. [NeurIPS'24].

Problem

• Maintaining dynamic point set diameter against adaptive adversaries
• Developing efficient k-center clustering for high-dimensional data
• Achieving robustness with poly-logarithmic update time guarantees

Innovation

• Dynamic algorithm maintains 2-approximate diameter efficiently
• Robust representative method with infrequent updates against adversaries
• Improved k-center clustering with poly-logarithmic update time