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Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

Anne Driemel, Jan Höckendorff, Ioannis Psarros, Christian Sohler, Di Yue

Abstract

Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm computes a $(2+ε)$-approximation for arbitrary $ε>0$ (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time $(1+ε)$-approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). 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 $X$ is doubling 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 $Y$ is doubling 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 $(k,\ell)$-median problem under discrete Fréchet distance when $\ell$ is constant. We further 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 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 (Talwar STOC 2004) to our setting. The running time of our algorithms is $2^{2^t} \tilde O(n+m)$ where $t=O(\mathrm{ddim} \log \frac{\mathrm{ddim}}ε)$ and where $\mathrm{ddim}$ is the doubling dimension of $X$ (resp.\ $Y$). The results also extend to the metric facility location problem.

Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

Abstract

Given a finite metric space the -median problem is to find a set of centers that minimizes . In general metrics, the best polynomial time algorithm computes a -approximation for arbitrary (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time -approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). We show that the -approximation algorithm can be generalized to the case when either or has bounded doubling dimension (but the other set not). The case when is doubling is motivated by the assumption that even though is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when is doubling 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 -median problem under discrete Fréchet distance when is constant. We further 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 has a bounded doubling dimension, we introduce a dimension reduction that replaces points from by sets of points in . To solve the case when has a bounded doubling dimension, we generalize Talwar's decomposition (Talwar STOC 2004) to our setting. The running time of our algorithms is where and where is the doubling dimension of (resp.\ ). The results also extend to the metric facility location problem.

Paper Structure

This paper contains 106 sections, 65 theorems, 271 equations, 2 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definitions
  3. Application: The $(k,\ell)$-Median Problem under Discrete Fréchet Distance
  4. Our Results
  5. Technical Overview
  6. Low-dimensional Centers
  7. Dimension reduction.
  8. Integrating the dimension reduction with portal-respecting distance.
  9. Dynamic program.
  10. Low-dimensional Clients
  11. Our new decomposition.
  12. Dynamic program.
  13. Applications to the $(k,\ell)$-Median Problem under Discrete Fréchet Distance
  14. Complexity reduction.
  15. Further Related Work
  16. ...and 91 more sections

Key Result

Theorem 1.3

There are randomized algorithms that, given as input $\varepsilon \in (0, \tfrac{1}{2})$, $n, m, k \in \mathbb{N}$ and $(X \cup Y, \mathop{\mathrm{\mathbf{d}}}\nolimits)$ with $|X| = n, |Y| = m, \mathop{\mathrm{\operatorname{ddim}}}\nolimits(Y) \leq \mathop{\mathrm{\operatorname{ddim}}}\nolimits$, c

Figures (2)

  • Figure 1: An illustration of the new decomposition $\mathcal{P}$. $C$ is a level $\ell$ cluster on $\mathcal{P}$, which has non-ornament child clusters $D_1, D_2, D_3, D_4$ and an ornament child $\{y\}$, where $y \in Y \setminus X$. The left figure shows how $C$ is partitioned. Black points are portals of $C$, and $\mathop{\mathrm{\mathbf{d}}}\nolimits(y, P_C) \leq 2 \sqrt{\rho} 2^\ell$. The right figure shows part of the decomposition $\mathcal{P}$, where circles are non-ornament nodes, and squares are ornaments.
  • Figure 2: Depicted are two time series $x$ and $y$ and the matched vertices through some traversal $T$. The traversal sectors of $(x,T)$ are $S^{(x,T)}_1 = \{x_1,x_2,x_3,x_4,x_5\}, S^{(x,T)}_2 =\{x_6,x_7,x_8\}$ and $S^{(x,T)}_3 = \{x_9,x_{10},x_{11}\}$. The orange highlighted vertices are the extrema values located in each traversal sector, which define the profile, i.e. the sequence $((x_2,x_4),(x_6,x_8),(x_{10},x_{11}))$ is the $3$-profile of $(x,T)$.

Theorems & Definitions (128)

  • Definition 1.1
  • Definition 1.2: ($k,\ell)$-median clustering problem
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 1.5
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: Doubling dimension GuptaKL03
  • Definition 2.2: Packing, covering and net
  • ...and 118 more