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

2025-03-28

AI Summary

This paper resolves the long-standing (1+ε)-spanner tree cover problem for doubling graphs: Does there exist a tree cover comprising only constantly many trees, with constant lightness, such that every vertex pair admits a (1+ε)-approximate shortest path in some tree? We present the first construction of a (1+ε)-spanner spanning tree cover with O(1) trees and O(1) lightness on doubling graphs—breaking the previously held belief that constant lightness is unattainable even for 2D point sets or general doubling metrics. Our approach integrates hierarchical clustering of doubling metrics, recursive construction of sparse spanning trees, labeled routing schemes, and distance oracle design. This yields three key results: (i) a light-weight tree cover with constant lightness and O(1) trees; (ii) a compact routing scheme using O(log n) bits per node; and (iii) a path-reporting distance oracle with O(1) query time and constant lookup latency.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 $Omega(log n)$; 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. [...]

Problem

• Constructs constant-stretch spanning tree covers for doubling graphs
• Resolves open question on constant lightness tree covers
• Provides compact routing schemes and distance oracles

Innovation

• Constant lightness spanning tree cover
• Compact routing scheme in labeled model
• Path-reporting distance oracle for doubling graphs