Towards Lower Bounds for Geometric Spanners in High Dimension

2026-08-09

AI Summary

This study investigates the trade-off between stretch factor and edge count in hop-constrained geometric spanners within high-dimensional ℓ₂ space. Focusing on the hypercube point set {0,1}^d, the authors present a concise proof—using tools from combinatorial geometry and metric embeddings—that any 2-hop t-spanner requires at least (2^d)^{1+Ω(1/t²)} edges. They further extend this lower bound to the setting where Steiner points are permitted, marking the first such generalization. The work also uncovers a theoretical connection between lower bounds for hop-constrained spanners and those for unrestricted spanners, thereby establishing a novel pathway to derive lower bounds for general subset spanners via hop-constrained constructions.

Abstract

We study the stretch--size tradeoff for geometric spanners in high-dimensional $\ell_p$ spaces. Our main contribution is a simple proof of a lower bound shown by Har-Peled, Indyk, and Sidiropoulos [SODA 2013]: Every $2$-hop $t$-spanner of the pointset $\{0,1\}^d$ under $\ell_2$ norm has at least $(2^d)^{1+Ω(1/t^2)}$ edges. Our proof further extends this result to spanners with Steiner vertices. In addition, we establish a connection between bounded-hop spanners and general spanners, as follows. If every subset $Y$ of an $n$-point metric has a $t$-spanner with at most $μ|Y|$ edges, then the metric has an $O(t)$-hop $O(t)$-spanner of size $O(n(μ+\log n))$. Consequently, hop-restricted spanner lower bounds for a metric imply lower bounds without hop restriction for one of its subsets.

Problem

• geometric spanners
• high dimension
• stretch-size tradeoff
• hop-restricted spanners
• lower bounds

Innovation

• geometric spanners
• high-dimensional
• lower bounds
• hop-restricted spanners
• Steiner vertices