Efficient Construction of Spanners in $d$-Dimensions

TL;DR

This paper presents an efficient method for constructing fault-tolerant geometric spanners in d-dimensional space with optimal bounds on degree, weight, and construction time, solving an open problem in the field.

Contribution

It introduces the first $O(n \,\log n)$ algorithms for constructing $k$-vertex fault-tolerant $t$-spanners with optimal bounds in $d$-dimensional space.

Findings

Constructs $t$-spanners with degree O(1) for $k=0$

Creates $k$-fault-tolerant $t$-spanners with degree O(k) for $k\ge 1$

Achieves optimal bounds on degree, weight, and construction time

Abstract

In this paper we consider the problem of efficiently constructing $k$-vertex fault-tolerant geometric $t$-spanners in $\dspace$ (for $k \ge 0$ and $t >1$). Vertex fault-tolerant spanners were introduced by Levcopoulus et. al in 1998. For $k=0$, we present an $O(n \log n)$ method using the algebraic computation tree model to find a $t$-spanner with degree bound O(1) and weight $O(\weight(MST))$. This resolves an open problem. For $k \ge 1$, we present an efficient method that, given $n$ points in $\dspace$, constructs $k$-vertex fault-tolerant $t$-spanners with the maximum degree bound O(k) and weight bound $O(k^2 \weight(MST))$ in time $O(n \log n)$. Our method achieves the best possible bounds on degree, total edge length, and the time complexity, and solves the open problem of efficient construction of (fault-tolerant) $t$-spanners in $\dspace$ in time $O(n \log n)$.