Towards a Unified Theory of Light Spanners I: Fast (Yet Optimal) Constructions

Seminal works on light spanners over the years provide spanners with optimal or near-optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Two shortcomings of all previous work on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly (e.g., some require large stretch and are thus suitable to general graphs, while others are naturally suitable to stretch $1 + \epsilon$). (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. This work aims at initiating a unified theory of light spanners by presenting a single framework that can be used to construct light spanners in a variety of graph classes. This theory is developed in two papers. The current paper is the first of the two -- it lays the foundations of the theory of light spanners and then applies it to design fast constructions with optimal lightness for several graph classes. Our new constructions are significantly faster than the state-of-the-art for every examined graph class; moreover, our runtimes are near-linear and usually optimal. Specifically, this paper includes the following results: (i) An $O(m \alpha(m,n))$-time construction of $(2k-1)(1+\epsilon)$-spanner with lightness $O(n^{1/k})$ for general graphs; (ii) An $O(n\log n)$-time construction of Euclidean $(1+\epsilon)$-spanners with lightness and degree both bounded by constants in the basic algebraic computation tree (ACT) model. This construction resolves a major problem in the area of geometric spanners, which was open for three decades; (iii) An $O(n\log n)$-time construction of $(1+\epsilon)$-spanners with constant lightness and degree, in the ACT model for unit disk graphs; (iv) a linear-time algorithm for constructing $(1+\epsilon)$-spanners with constant lightness for minor-free graphs.