Dependable Spanners via Unreliable Edges
Sariel Har-Peled, Maria C. Lusardi
TL;DR
This work investigates constructing dependable (1+ε)-spanners for point sets under independent random edge failures with survival probability ψ. It first resolves the 1D case by matching a clique-based deficiency bound with a near-linear edge-spanner, achieving a 1D exact spanner of size O(n/ψ · log n) and a deficiency of Θ(n/ψ · log(1/ψ)); it then introduces small-hop variants, notably a 4-hop exact spanner, using block decompositions and sparse bipartite connectors to reach size O(n/ψ^(4/3) · log n). The results extend to Rd by embedding the 1D construction into the locality-sensitive ordering framework of Chan et al., yielding a geometric spanner in Rd with near-linear size and high-probability short-hop connections for most pairs. Together, the paper demonstrates that near-linear-size dependable spanners are achievable even under massive edge failure and provides a pathway from 1D to high-dimensional geometric settings via LSOs and expander-based connectors.
Abstract
Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $\varepsilon,ψ\in (0,1)$ be parameters. Here, we consider the task of constructing a $(1+\varepsilon)$-spanner for $P$, where every edge might fail (independently) with probability $1-ψ$. For example, for $ψ=0.1$, about $90\%$ of the edges of the graph fail. Nevertheless, we show how to construct a spanner that survives such a catastrophe with near linear number of edges. The measure of reliability of the graph constructed is how many pairs of vertices lose $(1+\varepsilon)$-connectivity. Surprisingly, despite the spanner constructed being of near linear size, the number of failed pairs is close to the number of failed pairs if the underlying graph was a clique. Specifically, we show how to construct such an exact dependable spanner in one dimension of size $O(\tfrac{n}ψ \log n)$, which is optimal. Next, we build an $(1+\varepsilon)$-spanners for a set $P \subseteq \mathbb{R}^d$ of $n$ points, of size $O( C n \log n )$, where $C \approx 1/\bigl(\varepsilon^{d} ψ^{4/3}\bigr)$. Surprisingly, these new spanners also have the property that almost all pairs of vertices have a $\leq 4$-hop paths between them realizing this short path.