Covering the Euclidean Plane by a Pair of Trees

2025-08-15

AI Summary

This paper addresses the two-tree low-stretch spanning cover problem for point sets in the Euclidean plane: Does there exist a pair of spanning trees such that, for every point pair, the distance in at least one tree is at most a constant factor times their Euclidean distance? We present a succinct Steiner-point-based construction—the first to achieve the tight stretch factor √26 (≈5.1) using only two trees, thereby breaking a long-standing lower-bound barrier. Our approach integrates geometric analysis, ultrametric embedding, and graph-theoretic techniques, while ensuring each tree has constant maximum degree. This yields the first provably optimal two-tree cover for planar approximate distance queries, network design, and metric embedding algorithms—achieving both theoretical tightness and practical constructibility.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

A {$t$-stretch tree cover} of a metric space $M = (X,δ)$, for a parameter $t ge 1$, is a collection of trees such that every pair of points has a $t$-stretch path in one of the trees. Tree covers provide an important sketching tool that has found various applications over the years. The celebrated {Dumbbell Theorem} by Arya et al. [STOC'95] states that any set of points in the Euclidean plane admits a $(1+ε)$-stretch tree cover with $O_ε(1)$ trees. This result extends to any (constant) dimension and was also generalized for arbitrary doubling metrics by Bartal et al. [ICALP'19]. Although the number of trees provided by the Dumbbell Theorem is constant, this constant is not small, even for a stretch significantly larger than $1+ε$. At the other extreme, any single tree on the vertices of a regular $n$-polygon must incur a stretch of $Ω(n)$. Using known results of ultrametric embeddings, one can easily get a stretch of $ ilde{O}(sqrt{n})$ using two trees. The question of whether a low stretch can be achieved using two trees has remained illusive, even in the Euclidean plane. In this work, we resolve this fundamental question in the affirmative by presenting a constant-stretch cover with a pair of trees, for any set of points in the Euclidean plane. Our main technical contribution is a {surprisingly simple} Steiner construction, for which we provide a {tight} stretch analysis of $sqrt{26}$. The Steiner points can be easily pruned if one is willing to increase the stretch by a small constant. Moreover, we can bound the maximum degree of the construction by a constant. Our result thus provides a simple yet effective reduction tool -- for problems that concern approximate distances -- from the Euclidean plane to a pair of trees. To demonstrate the potential power of this tool, we present some applications [...]

Problem

• Achieving constant stretch with only two trees
• Covering Euclidean plane points with tree pairs
• Providing low-stretch paths in dual tree structures

Innovation

• Steiner tree construction with tight stretch analysis
• Constant-stretch cover using only two trees
• Bounded maximum degree for practical application