Light Spanners with Small Hop-Diameter

TL;DR

Addresses the optimal tradeoff between hop-diameter and lightness for spanners in doubling metrics by introducing a tree-cover framework that lifts tree-spanner guarantees to general metrics, yielding a $t$-spanner with hop-diameter $k$ and lightness $O(\gamma L k n^{2/k})$. The authors prove a matching lower bound for the $n$-point uniform line metric, showing the $O(k n^{2/k})$ exponent is tight for tree metrics and hence for doubling metrics. They also show how to obtain a $(1+\varepsilon)$-spanner in doubling spaces with hop-diameter $k$ and lightness $O(\varepsilon^{-O(d)}\cdot k n^{2/k})$, where $d$ is the doubling dimension. Together, these results provide a fine-grained, structural route to optimal spanner designs across all hop-diameter regimes.

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 $O(α_k(n))$, where $α_k$ 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 $γ$ trees, stretch $t$, and lightness $L$, then it also admits a $t$-spanner with hop-diameter $k$ and lightness $O(kn^{2/k}\cdot γL)$. 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 $k$.

Figures (2)

• Figure 1: An illustration of the proof of the lower bound for $k=2$ (\ref{['st:lb-2']}). There is a vertex $p \in M_1$ (highlighted in red) which is not incident on any edge going to $M_3$ or $M_4$. For every point $q \in M_4$, a $2$-hop path from $p$ to $q$ induces a long edge, highlighted in red.
• Figure 2: Monotonicity of the number of global points as we increase the interval size from $\ell_i$ to $\ell_{i+1}$. The points highlighted in red were global with respect to $\ell_i$ and became non-global with respect to $\ell_{i+1}$.

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.1: FL22
• Lemma 2.2
• proof
• Lemma 2.3
• Claim 2.4