A fast approximation scheme for low-dimensional k-means

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  • Vincent Cohen-Addad

We consider the popular k-means problem in ddimensional Euclidean space. Recently Friggstad, Rezapour, Salavatipour [FOCS'16] and Cohen-Addad, Klein, Mathieu [FOCS'16] showed that the standard local search algorithm yields a p1"q-approximation in time pn kq1-"Opdq, giving the first polynomial-time approximation scheme for the problem in low-dimensional Euclidean space. While local search achieves optimal approximation guarantees, it is not competitive with the state-of-the-art heuristics such as the famous kmeans++ and D2-sampling algorithms. In this paper, we aim at bridging the gap between theory and practice by giving a p1 "q-approximation algorithm for low-dimensional k-means running in time nk plog nqpd" 1qOpdq, and so matching the running time of the k-means++ and D2-sampling heuristics up to polylogarithmic factors. We speed-up the local search approach by making a non-standard use of randomized dissections that allows to find the best local move efficiently using a quite simple dynamic program. We hope that our techniques could help design better local search heuristics for geometric problems.

Original languageEnglish
Title of host publicationProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
EditorsArtur Czumaj
PublisherSociety for Industrial and Applied Mathematics
Publication date2018
ISBN (Electronic)9781611975031
Publication statusPublished - 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: 7 Jan 201810 Jan 2018


Conference29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
LandUnited States
ByNew Orleans
SponsorACM Special Interest Group on Algorithms and Computation Theory (SIGACT), SIAM Activity Group on Discrete Mathematics

ID: 203834632