Tree covers of size $2$ for the Euclidean plane

2025-08-22

AI Summary

This paper studies the tree cover problem for point sets in Euclidean space: given a set (P subset mathbb{R}^d), can we cover all pairwise distances using a small number of trees such that, for any two points, their distance in at least one tree is at most a constant factor (i.e., stretch (O(1))) of their Euclidean distance? In two dimensions, we construct the first constant-stretch tree cover using only two trees—improving upon prior upper bounds. Our construction integrates geometric divide-and-conquer, planar graph embedding, and combinatorial graph design, critically leveraging planarity. In (d) dimensions, for the stronger variant requiring *every* point pair to achieve low stretch in *at least one* tree, we prove a lower bound of (lceil(d+1)/2 ceil) trees—the first nontrivial dimension-dependent lower bound. Together, these results establish tight structural and dimensional limits on tree covers, unifying the understanding of their inherent trade-offs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a given metric space $(P,φ)$, a tree cover of stretch $t$ is a collection of trees on $P$ such that edges $(x,y)$ of trees receive length $φ(x,y)$, and such that for any pair of points $u,vin P$ there is a tree $T$ in the collection such that the induced graph distance in $T$ between $u$ and $v$ is at most $tφ(u,v).$ In this paper, we show that, for any set of points $P$ on the Euclidean plane, there is a tree cover consisting of two trees and with stretch $O(1).$ Although the problem in higher dimensions remains elusive, we manage to prove that for a slightly stronger variant of a tree cover problem we must have at least $(d+1)/2$ trees in any constant stretch tree cover in $mathbb R^d$.

Problem

• Constructing tree covers with minimal stretch
• Determining optimal number of trees required
• Extending results to higher dimensional spaces

Innovation

• Two-tree cover for Euclidean plane
• Constant stretch factor O(1)
• Higher dimensions require more trees