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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.80
Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

Authors: Anne Driemel, Jan Höckendorff, Ioannis Psarros, Christian Sohler, and Di Yue

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
In the metric k-median problem we are given a finite metric space (X∪ Y, 𝐝) and the objective is to compute a set of k centers C ⊆ Y that minimizes ∑_{p ∈ X} min_{c ∈ C} 𝐝(p,c). In general metric spaces, the best polynomial time algorithm, which is due to Cohen-Addad, Grandoni, Lee, Schwiegelshohn, and Svensson [Vincent Cohen-Addad et al., 2025], computes a (2+ε)-approximation for arbitrary constant ε > 0. However, if the metric space has bounded doubling dimension, a near linear time (1+ε)-approximation algorithm is known due to the work of Cohen-Addad, Feldmann, and Saulpic [Vincent Cohen{-}Addad et al., 2021]. In this paper, we show that the (1+ε)-approximation algorithm can be generalized to the case when either X or Y has bounded doubling dimension (but the other set not). The case when X has bounded doubling dimension is motivated by the assumption that even though X is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when Y has bounded doubling dimension is perhaps more natural. It is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the (k,𝓁)-median problem under discrete Fréchet distance when 𝓁 is constant. The latter problem is a version of the k-median problem under Fréchet distance when the input consists of time series of z reals and where the centers are time series of 𝓁 reals [Anne Driemel et al., 2016]. Previously, for this problem no (1+ε)-approximation algorithm with running time polynomial in k was known. We also introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fréchet distance. In order to solve the case when Y has a bounded doubling dimension, we introduce a form of dimension reduction that replaces points from X by sets of points in Y. To solve the case when X has a bounded doubling dimension, we generalize Talwar’s decomposition [Kunal Talwar, 2004] of doubling metrics to our setting. The running time of our algorithms is 2^{2^t} Õ(n+m) where t = O(ddim log ddim/ε) and where ddim is the doubling dimension of X (resp. Y). The results also extend to the metric (uncapacitated) facility location problem. We believe that our techniques are likely applicable to other problems.
Cite as

Anne Driemel, Jan Höckendorff, Ioannis Psarros, Christian Sohler, and Di Yue. Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{driemel_et_al:LIPIcs.ICALP.2026.80,
  author =	{Driemel, Anne and H\"{o}ckendorff, Jan and Psarros, Ioannis and Sohler, Christian and Yue, Di},
  title =	{{Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{80:1--80:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.80},
  URN =		{urn:nbn:de:0030-drops-264693},
  doi =		{10.4230/LIPIcs.ICALP.2026.80},
  annote =	{Keywords: Approximation Algorithms, Doubling Spaces, Facility Location, k-Median, Discrete Fr\'{e}chet Distance}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.82
Dimension Reduction for Clustering: The Curious Case of Discrete Centers

Authors: Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of n points into dimension O(log n) with low distortion of their distances. This dimension bound is tight in general, but one can bypass it for specific problems. Indeed, tremendous progress has been made for clustering problems, especially in the continuous setting where centers can be picked from the ambient space ℝ^d. Most notably, for k-median and k-means, the dimension bound was improved to O(log k) [Makarychev, Makarychev and Razenshteyn, STOC 2019]. We explore dimension reduction for clustering in the discrete setting, where centers can only be picked from the dataset, and present two results that are both parameterized by the doubling dimension of the dataset, denoted as ddim. The first result shows that dimension O_{ε}(ddim + log k + log log n) suffices, and is moreover tight, to guarantee that the cost is preserved within factor 1±ε for every set of centers. Our second result eliminates the log log n term in the dimension through a relaxation of the guarantee (namely, preserving the cost only for all approximately-optimal sets of centers), which maintains its usefulness for downstream applications. Overall, we achieve strong dimension reduction in the discrete setting, and find that it differs from the continuous setting not only in the dimension bound, which depends on the doubling dimension, but also in the guarantees beyond preserving the optimal value, such as which clusterings are preserved.
Cite as

Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue. Dimension Reduction for Clustering: The Curious Case of Discrete Centers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 82:1-82:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jiang_et_al:LIPIcs.ITCS.2026.82,
  author =	{Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Sapir, Shay and Silwal, Sandeep and Yue, Di},
  title =	{{Dimension Reduction for Clustering: The Curious Case of Discrete Centers}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{82:1--82:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.82},
  URN =		{urn:nbn:de:0030-drops-253698},
  doi =		{10.4230/LIPIcs.ITCS.2026.82},
  annote =	{Keywords: dimension reduction, clustering, k-median, k-means, doubling dimension}
}
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