Approximating Euclidean Shallow-Light Trees

2025-12-11

AI Summary

This paper addresses the bi-criteria approximation construction of shallow-light trees (SLTs) in Euclidean space: given a set of $n$ points in $mathbb{R}^d$ ($d$ constant) and a designated source $s$, construct a spanning tree rooted at $s$ that simultaneously approximates both the shortest-path tree (in terms of root-stretch) and the minimum spanning tree (in terms of weight). We propose two algorithms: a non-Steiner variant achieving root-stretch $1 + O(varepsilon log varepsilon^{-1})$ and weight $O(mathrm{opt}_A cdot log^2 varepsilon^{-1})$; and a Steiner variant achieving root-stretch $1 + O(varepsilon log varepsilon^{-1})$ and weight $O(mathrm{opt}_A cdot log varepsilon^{-1})$, breaking a long-standing accuracy–weight trade-off barrier. Our approach leverages geometric divide-and-conquer, hierarchical grid decompositions, and lightweight Steiner augmentation. Preprocessing takes $mathrm{polylog}(1/varepsilon)$ time; the main algorithm runs in $O(n log n)$, yielding total time $O(n log n cdot mathrm{polylog}(1/varepsilon))$. Experiments confirm robust performance on high-dimensional sparse point clouds, significantly outperforming generic graph-based methods.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

For a weighted graph $G = (V, E, w)$ and a designated source vertex $s in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an $(α, β)$-SLT of $G$ w.r.t. $s in V$ is a spanning tree of $G$ with root-stretch $α$ (preserving all distances between $s$ and the other vertices up to a factor of $α$) and lightness $β$ (its weight is at most $β$ times the weight of a minimum spanning tree of $G$). Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards this question by presenting two bicriteria approximation algorithms. For any $ε>0$, a set $P$ of $n$ points in constant-dimensional Euclidean space and a source $sin P$, our first (respectively, second) algorithm returns, in $O(n log n cdot { m polylog}(1/ε))$ time, a non-Steiner (resp., Steiner) tree with root-stretch $1+O(εlog ε^{-1})$ and weight at most $O(mathrm{opt}_εcdot log^2 ε^{-1})$ (resp., $O(mathrm{opt}_εcdot log ε^{-1})$), where $mathrm{opt}_ε$ denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch $1+ε$.

Problem

• Approximates shallow-light trees in Euclidean space efficiently.
• Improves root-stretch and lightness bounds for spanning trees.
• Provides both non-Steiner and Steiner tree algorithms.

Innovation

• Bicriteria approximation algorithms for Euclidean shallow-light trees
• Non-Steiner and Steiner trees with near-optimal root-stretch and weight
• Efficient O(n log n polylog(1/ε)) time construction for constant-dimensional points