Coresets for Clustering in Graphs of Bounded Treewidth

07/10/2019
by   Vladimir Braverman, et al.
0

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems (on graph metrics) are essential to data analysis and used for example in road networks and data visualization. Specifically, we consider (k, z)-Clustering, where given a metric space (V, d), the goal is to minimize, over all k-point center sets C, the objective ∑_x ∈ Vd^z(x, C). This problem is a well-known generalization of both k-Median (z=1) and k-Means (z=2). A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set. Coresets offer significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset of size O_ϵ, k, z(tw(G)) for (k, z)-Clustering in a graph G whose treewidth is tw(G). The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of O(tw(G)) on the shattering dimension under any point weights. Previously, the only construction applicable to graph metrics, even for z=1, was a generic one with size O_ϵ, k( n) where n=|V| [Feldman and Langberg, STOC 2011]. We complement our construction with an Ω_ϵ, k(tw(G)) size lower bound, which matches our construction's linear dependence on tw(G). This further provides the first proof that the O( n) factor in the generic upper bound is indeed necessary, and also justifies restricting the graph topology.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/14/2020

Coresets for Clustering in Euclidean Spaces: Importance Sampling is Nearly Optimal

Given a collection of n points in ℝ^d, the goal of the (k,z)-clustering ...
research
03/02/2023

Coresets for Clustering in Geometric Intersection Graphs

Designing coresets–small-space sketches of the data preserving cost of t...
research
04/07/2018

ε-Coresets for Clustering (with Outliers) in Doubling Metrics

We study the problem of constructing ε-coresets for the (k, z)-clusterin...
research
04/16/2020

Coresets for Clustering in Excluded-minor Graphs and Beyond

Coresets are modern data-reduction tools that are widely used in data an...
research
04/06/2023

Parameterized Approximation Schemes for Clustering with General Norm Objectives

This paper considers the well-studied algorithmic regime of designing a ...
research
09/04/2018

Relaxed Voronoi: a Simple Framework for Terminal-Clustering Problems

We reprove three known algorithmic bounds for terminal-clustering proble...
research
03/11/2019

Coresets for Ordered Weighted Clustering

We design coresets for Ordered k-Median, a generalization of classical c...

Please sign up or login with your details

Forgot password? Click here to reset