Title: Dynamic Diameter in High-Dimensions against Adaptive Adversary and Beyond
Venue: NeurIPS 2025
Accepted: yes
Keywords: Dynamic algorithm

diameter problem

adaptive adversary

high dimensional space
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 \emph{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 \emph{worst-case} update time of $poly(d, \log n)$, 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 \emph{simultaneously} achieves a $2$-approximation guarantee and robustness against an adaptive adversary. We also give an improved dynamic $(4+\epsilon)$-approximation algorithm for the $k$-center problem, also resilient to an adaptive adversary. Our clustering algorithm achieves an amortized update time of $k^{2.5} d \cdot poly(\epsilon^{-1}, \log n)$, improving upon the amortized update time of $k^6 d \cdot poly( \epsilon^{-1}, \log n)$ by Biabani et al. [NeurIPS'24].


Peer reviews:
comments: SUMMARY:
The paper tackles two classical geometric problems—maintaining the diameter and $k$-center clustering—over fully dynamic point sets in high-dimensional Euclidean space, under the strongest “randomness-adaptive” adversary model. 

The main contributions are
1. a 2-approximation algorithm for dynamic diameter (and minimum enclosing ball) with worst-case update time $O\bigl(d^{5}\log^{1.5}d\log^{1.5}(n/\delta)\bigr)$, and
1. a $(4+\varepsilon)$-approximation for fully-dynamic $k$-center with amortized update time $k^{2.5}d\cdot \mathrm{poly}(\varepsilon^{-1},\log n)$, improving the previous $k^6\cdot \mathrm{poly}(d, \varepsilon^{-1}, \log n)$ bound. 

Both results hinge on a novel use of $\varepsilon$-robust centerpoints (Tukey-depth ≥ $\Omega(1/d^2)$) to pick "representative" anchors that survive many deletions.
The diameter algorithm also employs a careful de-amortization step.

STRENGTHS AND WEAKNESSES:
### Strengths
1. Fully dynamic algorithms for the diameter problem against adaptive adversaries with any non-trivial
approximation and worst-case $\mathrm{poly}(d \log n)$ update time were previously not known.
1. The improvement in update time for fully dynamic, adaptive $k$-center seems significant.
1. The robust centerpoint idea is clean and may be applicable in other settings.

---

### Weaknesses
1. The $d^5$ update time may be impractical for high-dimensional settings.

QUESTIONS:
N/A

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents two fully dynamic algorithms:\n* dynamic maintenance of a 2-approximation to diameter in O(d^5 log^3 n) worst case update time\n* dynamic maintenance of (4+epsilon)-approximate k-center in O(k^2.5 d poly(epsilon^-1, log n)) amortized update time\nHere, d is the dimension of the underlying Euclidean space, and n is the current number of points.\n\nBoth results hold against a strong notion of adaptive adversaries, called randomness-adaptive adversary: the adversary has access to the algorithm, all random choices and outputs of the algorithm up to that point. \n\nThe diameter result relies on finding an epsilon-robust representative point of pointset P - intuitively, a point x such that dist(x, P) stays a 2 approximation to diam(P) even after any epsilon*n changes to set P. Finding such a point guarantees that we do not need to recompute the representative (and hence the diameter) of the set too often, and that the algorithm is also independent of the exact updates chosen by the adversary. The computation of the representative point is reduced to a black-box algorithm for finding an alpha-centerpoint (for alpha=1/poly d). After epsilon*n deletions to P, we simply recompute the representative point from scratch. The deamortization of this algorithm uses a standard technique by Motwani and Raghavan (1999).\n\nThe k-center algorithm does the following for each “guess” (1+epsilon)^l of the size of the optimal solution OPT(P). In the first round, we sample some points from P, and choose a point out of these that has most of the remaining points at distance at most 2(1+epsilon)^l. The chosen center, together with its closest points, is then deleted from P, and we recurse on the remaining points. Intuitively, if our “guess” of OPT(P) was good, we only have k rounds, and hence choose k centers.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper is well written, easy to follow, and provides good intuition of the techniques used. Even simple results are proven in detail in the appendix, making the whole paper self-contained.\n* The literature overview is comprehensive, and the summary of techniques in other papers clearly highlights the contributions of this work.\n* The results hold for a strong notion of adaptive adversaries: the adversary has access to the algorithm, all random choices, and outputs of the algorithm up to that point. \n\nWeaknesses: \n* While the diameter result uses some novel techniques, I think paying a large polynomial dependency on dimension just for getting worst-case guarantees feels like overkill to me. \n* The k-center result is less novel technically, but shows a clear improvement in the dependency on the number of centers. To me, it would be interesting to get such a result for the 2+eps approx regime, which is known to be tight for this problem.\nSome minor comments (see Questions section).\n\nQUESTIONS:\nIs there a possibility of deamortizing the k-center algorithm? I assume that the recursive nature of the algorithm might make this hard, but maybe one can make these rounds of choosing the centers less dependent on each other.\n\nSome minor comments about the presentation: \nTable 1 shows dependencies of d, while Table 2 doesn’t, making it a bit inconsistent.\nLine 65: these performance measures also apply to the diameter problem.\nLine 128: citation format issue\nLine 201: who see -> who sees\nLine 233: should be P\\in R^d\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper is about fully dynamic (allowing insertions & deletions) constant factor approximation algorithms for the diameter and k-center clustering of the input. \n\nMoreover the algorithms are robust against adversaries that have full knowledge of the past decisions and random bits used by the algorithm, which is argued to be the strongest model that admits constant factor approximations.\n\nThe paper combines in a clever way, simple online algorithm with recent advances in computational geometry and with classical robust statistics.\n\nthe algorithm for maintaining the diameter is super simple and uses in a clever way the robustness of so called Tukey centerpoints, i.e., points that lie *deeply* in the interior of the point set:\n\nAny point (not necessarily in the input) in the convex hull (CH) of the current set of points is a 2-approximation for the diameter (or 1-center). The computed centerpoint lies deeply in the CH and has the property that at least d^2 many points need to be removed before it may fall out of the CH.\n\nA caveat is that such points are relatively expensive to compute, in roughly d^7. But once computed, they may not fall out of the CH on insertions and since they remain valid 2-approximations for at least d^2 deletions. Thus, the amortized time is only about d^5. The calculations steps are additionally cleverly distributed over all steps to additionally de-amortize the algorithm.\n\nThe contributions are well compared to prior work in two tables covering dynamic k-center and diameter problems, respectively.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nthe paper is well written, and contains a good presentation of related work and comparisons to prior results in similar settings.\n\nsimplicity of the algorithms\n\nclever combination of algorithmic concepts with computational geometry and basic robust statistics.\n\n \nWeakness:\nThe update time is rather high, and even though previous algorithms had other caveats, and in particular did not work against adaptive adversaries, they had achieved $1+\\epsilon$ or $\\sqrt{2}$ approximation which is better than the presented 2-approx.\n\nQUESTIONS:\n* to what extent do you think can the approximation factor be improved? Any comments?\n\n* to what extent do you think can the update time be improved? Why do you state O(k^6) as being \"prohibitively large\" even for small $k$ for k-center while O(d^5) is good enough for the diameter?\n\n* for k-center, any ideas on reaching linear k dependence as in prior work while dealing with an adaptive adversary?\n\n* I'd like to see a more formal definition of the Definition 2.4 (Tukey Depth). The current one reads more like an informal description.\n\n* I'd like to see a citation of Tukey's original work. Some more or less recent contributions from the computational geometry community are present, but the classical statistical work should also be cited. Especially for such an important notion named after its inventor.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors study the problem of maintaining the diameter and k-center clustering under dynamic updates (addition/deletion of points) under the presence of an adaptive adversary. The authors propose fully dynamic algorithms that achieve a 2-approximation for the diameter problem and a $(4 + \\epsilon)$-approximation for the k-center clustering problem. A simple 2-approximation dynamic algorithm picks a random anchor point $p \\in P$ and maintains its furthest neighbor throughout updates, remains the best known in terms of both approximation and update time. In this work, the authors provide a significant improvement, by designing a 2-approximation algorithm with worst case update of $O(d^5)$ under adaptive adversaries. Their work introduces novel techniques such as robust representatives based on center points and de-amortization strategies to ensure worst-case update time guarantees.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n\nThe theoretical claims are strong (improvements over both the approximation guarantee and worst-case update time) and the algorithm relies on highly non-trivial techniques. \n\nThe guarantees hold against an adaptive adversary that also has access to randomness of the algorithm, which is a significantly harder case.\n\n**Weaknesses**:\n\nThe authors heavily rely on a prior work to find the center point. Essentially these center points serve as the anchor points (from folklore).\n\nThe authors explain the role of adaptive adversaries, motivating it. Such an adversary model is common in the literature, and the writing could be improved to focus more on the overview of technical challenges.\n\nQUESTIONS:\nIt would have been great, if the authors highlighted the intution behind their novel sampling to improve update time.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}