The k-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the k-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is O(log k)-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in k-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of O(log² k). In this paper, we close the gap by providing a tight, O(log k)-approximate guarantee for the k-means++ algorithm with noise.
@InProceedings{grunau_et_al:LIPIcs.ESA.2023.55, author = {Grunau, Christoph and \"{O}z\"{u}do\u{g}ru, Ahmet Alper and Rozho\v{n}, V\'{a}clav}, title = {{Noisy k-Means++ Revisited}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {55:1--55:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.55}, URN = {urn:nbn:de:0030-drops-187080}, doi = {10.4230/LIPIcs.ESA.2023.55}, annote = {Keywords: clustering, k-means, k-means++, adversarial noise} }
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