Optimal Euclidean Tree Covers

TL;DR

This work studies optimal Euclidean tree covers: a collection of trees on a point set in $\,\mathbb{R}^d$ that guarantees a $(1+\varepsilon)$-stretch path for every pair of points in some tree, with improved bounds on the number of trees and the maximum degree. The authors introduce a two-stage approach that first reduces the problem to a single level and then handles that level with a refined geometric decomposition into strips and major/minor partitions, replacing previous packing-based barriers with conic partitioning to achieve $O_d(\varepsilon^{-d+1}\log(1/\varepsilon))$ trees and absolute constant degree per point (up to polylog factors). They also present a Steiner-tree cover achieving $O_d(\varepsilon^{(-d+1)/2}\log(1/\varepsilon))$ trees and provide matching (up to $\log(1/\varepsilon)$ factors) lower bounds, along with linear-time construction in the number of edges. The work extends to higher dimensions, yields constant-degree constructions, and yields routing schemes in low-dimensional Euclidean spaces with compact labels and sublinear routing tables. Overall, the paper tightens the fundamental trade-off between tree cover size, degree, and stretch, and enables efficient routing and distance-preserving structures in Euclidean spaces.

Abstract

A $(1+\varepsilon)\textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated $\textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $\textit{maximum degree}$ of a point in the tree cover may be as large as $Ω(\log Φ)$, where $Φ$ is the aspect ratio of the input point set. In this work we present a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$ trees, which is optimal (up to the $\log(1/\varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $\textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+\varepsilon)$-stretch $\textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n \log n)$ term; this improves over the running time underlying the Dumbbell Theorem.

Figures (3)

• Figure 1: A major strip partition (in blue) in direction $\theta$, and a minor strip partition (in purple) in direction $\theta^\bot$. Points $x$ and $y$, and the vector $v$ broken into components parallel to and orthogonal to $\theta$.
• Figure 2: Point sets $A$ and $B$, both in the same major strip (blue) but in different minor strips (purple). The points $a, a^*$, and $b$, with $\operatorname{score}_\theta(a) \le \operatorname{score}_\theta(a^*) \le \operatorname{score}_\theta(b)$, and the line $L$ passing through $a^*$.
• Figure 3: The binary trees $T_A$ and $T_B$, constructed greedily from point sets $A$ and $B$

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Cf. Cha98GH23
• Definition 2.2
• Definition 2.3
• Lemma 2.4: Reduction to partial tree cover
• proof
• Claim 2.5
• proof