A Fast and Simple $(1+ε)$-Approximation for Minimum Spanning Trees in Doubling Metrics

2026-07-14

AI Summary

This work proposes a deterministic $(1+\varepsilon)$-approximation algorithm for the minimum spanning tree (MST) problem in doubling metric spaces, significantly improving the dependence of the running time on the accuracy parameter $\varepsilon$. By integrating metric decomposition based on doubling dimension, hierarchical clustering, and sparse graph construction, the algorithm reduces the $\varepsilon$-dependence from the previously known $\varepsilon^{-O(\mathrm{ddim})}$ to nearly linear, namely $\varepsilon^{-1}$. The approach also reveals structural properties of MSTs in such spaces, particularly concerning vertex degrees. When the doubling dimension $\mathrm{ddim}$ is bounded, the algorithm runs in $2^{O(\mathrm{ddim})} n (\log n + \varepsilon^{-1} \log^4(1/\varepsilon))$ time, achieving an almost $\varepsilon^{-1}$-fold speedup over the best prior deterministic algorithm in Euclidean space.

Abstract

The minimum spanning tree (MST) problem is one of the most basic optimization problems on metric spaces and graphs. We study the problem of computing a $(1+ε)$-approximation to the MST of an $n$-point metric space $(X, \mathbf{d})$ of doubling dimension $\mathrm{ddim}$. In doubling metrics, previous deterministic algorithms incur a running time with dependence $ε^{-O(\mathrm{ddim})}$. We give a deterministic algorithm that computes a $(1+ε)$-approximation to MST in time $2^{O(\mathrm{ddim})} n \bigl(\log n + ε^{-1} \log^4(1/ε)\bigr)$. For bounded doubling dimension, this improves the previous dependence on $ε$ from $ε^{-O(\mathrm{ddim})}$ to essentially linear in $ε^{-1}$. Moreover, as a special case, our result improves the previous best deterministic running time for bounded-dimensional Euclidean metrics due to Arya and Mount~[SODA'16] by almost a factor of $ε^{-1}$. We also show that, unlike in bounded-dimensional Euclidean spaces, MSTs in bounded doubling metrics can have arbitrarily large maximum degree, while every doubling metric nevertheless admits a $(1+ε)$-approximate MST of maximum degree $2^{O(\mathrm{ddim})}\log(1/ε)$.

Problem

• minimum spanning tree
• doubling metrics
• approximation algorithm
• doubling dimension
• computational complexity

Innovation

• doubling metrics
• minimum spanning tree
• approximation algorithm
• deterministic algorithm
• doubling dimension