• Corpus ID: 245131065

# Optimal Fully Dynamic k-Centers Clustering

@article{Bateni2021OptimalFD,
title={Optimal Fully Dynamic k-Centers Clustering},
author={MohammadHossein Bateni and Hossein Esfandiari and Rajesh Jayaram and Vahab S. Mirrokni},
journal={ArXiv},
year={2021},
volume={abs/2112.07050}
}
• Published 13 December 2021
• Computer Science
• ArXiv
We present the first algorithm for fully dynamic k-centers clustering in an arbitrary metric space that maintains an optimal 2 + ǫ approximation in O(k · polylog(n,∆)) amortized update time. Here, n is an upper bound on the number of active points at any time, and ∆ is the aspect ratio of the data. Previously, the best known amortized update time was O(k · polylog(n,∆)), and is due to Chan, Gourqin, and Sozio [CGS18]. We demonstrate that the runtime of our algorithm is optimal up to polylog(n…

## References

SHOWING 1-10 OF 61 REFERENCES
On coresets for k-means and k-median clustering
• Computer Science
STOC '04
• 2004
This paper shows the existence of small coresets for the problems of computing k-median/means clustering for points in low dimension, and improves the fastest known algorithms for (1+ε)-approximate k-means and k- median.
Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time
• Computer Science
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
The first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph---which undergoes both edge insertions and deletions---in polylogarithmic time is presented and a simpler variant of the algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.
Consistent k-Clustering for General Metrics
• Computer Science
SODA
• 2021
This work shows how to maintain a constant-factor approximation for the $k-median problem by performing an optimal (up to polylogarithimic factors) number$\widetilde{O}(k)$of center swaps. Streaming Algorithms for k-Center Clustering with Outliers and with Anonymity • Computer Science APPROX-RANDOM • 2008 This work develops the first streaming algorithms achieving a constant-factor approximation to the cluster radius for two variations of the k-center clustering problem, which is a common problem in the analysis of large data sets. Optimal Time Bounds for Approximate Clustering • Computer Science Machine Learning • 2004 Using successive sampling, an algorithm is developed for the k-median problem that runs in O(nk) time for a wide range of values of k and is guaranteed, with high probability, to return a solution with cost at most a constant factor times optimal. Approximation algorithms for np -hard clustering problems • Computer Science • 2002 This dissertation studies the classic facility location and k-median problems in the context of clustering, and formulate and study a new optimization problem that is called the online median problem, which a natural generalization of the greedy strategy that is hierarchically greedy yields an algorithm. Consistent k-Clustering • Computer Science ICML • 2017 A lower bound is proved, showing that Ω(k log n) changes are necessary in the worst case for a wide range of objective functions, and an algorithm is given that needs onlyO(log n)Changes to maintain a constant competitive solution. Fully Dynamic Consistent Facility Location • Computer Science NeurIPS • 2019 The cost of the solution maintained by the algorithm at any time is very close to the cost of a solution obtained by quickly recomputing a solution from scratch at time$t$while having a much better running time. Fully dynamic hierarchical diameter k-clustering and k-center • Computer Science ArXiv • 2019 At any point of time, with probability at least$1-1/n$, the data structure can correctly answer all queries for cluster representatives in$O(d \ell \log n \log \Delta)\$ time per query.
Fully Dynamic Maximal Independent Set with Sublinear in n Update Time
• Computer Science, Mathematics
SODA
• 2019
This paper presents the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an Õ( √ n) expected amortized update time algorithm.