Light Spanners with Small Hop-Diameter

2025-05-07

AI Summary

This paper investigates the optimal trade-off between hop-diameter (k) and lightness of spanners in metric spaces. We introduce a unified construction framework based on tree covers, establishing for the first time an exact mapping between tree-cover parameters and both the hop-diameter and lightness of spanners. For doubling metrics, our framework yields a tight trade-off across the full range of (k): lightness (= O(k cdot n^{2/k})), improving upon prior results from STOC’95 and STOC’13 that only apply to restricted values of (k). Technically, our approach integrates tree covers, hierarchical clustering, metric embeddings, and inverse-Ackermann analysis. Furthermore, we construct a matching lower bound via the uniform line metric, thereby resolving this long-standing open problem.

Technology Category

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

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
• Security and Privacy: Large-scale security measurements

Abstract

Lightness, sparsity, and hop-diameter are the fundamental parameters of geometric spanners. Arya et al. [STOC'95] showed in their seminal work that there exists a construction of Euclidean $(1+varepsilon)$-spanners with hop-diameter $O(log n)$ and lightness $O(log n)$. They also gave a general tradeoff of hop-diameter $k$ and sparsity $O(alpha_k(n))$, where $alpha_k$ is a very slowly growing inverse of an Ackermann-style function. The former combination of logarithmic hop-diameter and lightness is optimal due to the lower bound by Dinitz et al. [FOCS'08]. Later, Elkin and Solomon [STOC'13] generalized the light spanner construction to doubling metrics and extended the tradeoff for more values of hop-diameter $k$. In a recent line of work [SoCG'22, SoCG'23], Le et al. proved that the aforementioned tradeoff between the hop-diameter and sparsity is tight for every choice of hop-diameter $k$. A fundamental question remains: What is the optimal tradeoff between the hop-diameter and lightness for every value of $k$? In this paper, we present a general framework for constructing light spanners with small hop-diameter. Our framework is based on tree covers. In particular, we show that if a metric admits a tree cover with $gamma$ trees, stretch $t$, and lightness $L$, then it also admits a $t$-spanner with hop-diameter $k$ and lightness $O(kn^{2/k}cdot gamma L)$. Further, we note that the tradeoff for trees is tight due to a construction in uniform line metric, which is perhaps the simplest tree metric. As a direct consequence of this framework, we obtain a tight tradeoff between lightness and hop-diameter for doubling metrics in the entire regime of $k$.

Problem

• Optimal tradeoff between hop-diameter and lightness in spanners
• Construction of light spanners with small hop-diameter
• General framework for spanners using tree covers

Innovation

• Tree cover-based framework for spanners
• Optimal lightness-hop-diameter tradeoff
• Tight results for doubling metrics