Light Tree Covers, Routing, and Path-Reporting Oracles via Spanning Tree Covers in Doubling Graphs

TL;DR

This paper addresses the problem of constructing tree-based sparsifiers with both strong distance preservation and structural simplicity in doubling graphs. It introduces a spanning $(1+\varepsilon)$-tree cover built from a strong-diameter hierarchical partition family and a novel preservable-set framework, achieving a constant number of trees and $(1+\varepsilon)$ stretch without introducing Steiner edges. By leveraging the spanning-tree cover, the authors derive a light tree cover, a compact routing scheme with near-optimal label/table sizes, and a path-reporting distance oracle that attains linear-space bounds for constant doubling dimension. The work unifies tree covers and spanners in doubling metrics to yield practical, near-optimal routing and distance-answering tools, with potential broad impact on network design and geometric data processing. Overall, it provides new techniques for constructing constant-stretch spanning tree covers with applications that were previously out of reach for spanning structures in general graphs.

Abstract

A $(1+\varepsilon)$-stretch tree cover of an edge-weighted $n$-vertex graph $G$ is a collection of trees, where every pair of vertices has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated Dumbbell Theorem by 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 a constant number of trees, where the constant depends on $\varepsilon$ and the dimension $d$. This result was generalized for arbitrary doubling metrics by Bartal et. al. [ICALP'19]. While the total number of edges in the tree covers of Arya et. al. and Bartal et. al. is $O(n)$, all known tree cover constructions incur a total lightness of $Ω(\log n)$; whether one can get a tree cover of constant lightness has remained a longstanding open question, even for 2-dimensional point sets. In this work we resolve this fundamental question in the affirmative, as a direct corollary of a new construction of $(1+\varepsilon)$-stretch spanning tree cover for doubling graphs; in a spanning tree cover, every tree may only use edges of the input graph rather than the corresponding metric. To the best of our knowledge, this is the first constant-stretch spanning tree cover construction (let alone for $(1+\varepsilon)$-stretch) with a constant number of trees, for any nontrivial family of graphs. Concrete applications of our spanning tree cover include a $(1+\varepsilon)$-stretch light tree cover, a compact $(1+\varepsilon)$-stretch routing scheme in the labeled model, and a $(1+\varepsilon)$-stretch path-reporting distance oracle, for doubling graphs. [...]

Figures (3)

• Figure 1: (a) A tree obtained from a hierarchy $\mathbb{P}^i \in \mathfrak{P}$ by connecting the representative $x_C$ of every cluster $C$ to those of its children. In this example, $x_C$ has 9 children, but we only show 4 for a clearer picture. (b) A $(1+\varepsilon)$-stretch path from $u$ to $v$; clusters $C_{i-1}(u)$ and $C_{i-1}(v)$ have diameter roughly $\varepsilon \cdot d_G(u,v)$.
• Figure 2: (a) The spanning tree $T_{\widehat{C}}$ for $\widehat{C}$ has to preserve both the subpath $\pi_C$ inside $\widehat{C}$, as well as the shortest path $\pi_{\widehat{C}}$ associated with $\widehat{C}$. (b) A sketch graph $H$ with respect to preservable set $\mathcal{P}$ (the orange paths; this includes the highway $\pi$ of $C$, and the path between the preserving pair of $C$ belonging to shaded clusters), inter-cluster edges $I$ (the bold red edges), and fake edges (the thin blue edges).
• Figure 4: $\mathcal{C}[\pi]$ and $\mathcal{C}[P_{xy}]$ intersect. In this case, we add $P_{xy}[x_1:x_{j_3 - 1}]$ and $P_{xy}[x_{j_4 + 1:x_k}]$ to $\mathcal{P}$. The left figure indicates the case that $x_{j_4}$ (the vertex before $x_{j_4 + 1}$ in $P_{xy}$) is in some cluster in $\mathcal{C}[\pi]$. The right figure indicates the case that $x_{j_4}$ is in some cluster in $\mathcal{C}[P_{xy}[x_1:x_{j_3 - 1}]]$.

Theorems & Definitions (54)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Theorem 1.5
• Theorem 1.6
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.2