Towards a Unified Theory of Light Spanners I: Fast (Yet Optimal) Constructions (2106.15596v5)

Abstract: Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. Our framework is developed in two papers. The current paper is the first of the two -- it lays the basis of the unified framework and then applies it to design fast constructions with optimal lightness for several graph classes. Among various applications and implications of our framework, we highlight here the following: _ In low-dimensional Euclidean spaces, we present an $O(n\log n)$-time construction of $(1+\epsilon)$-spanners with lightness and degree both bounded by constants in the algebraic computation tree (ACT). Our construction resolves a major problem in the area of geometric spanners, which was open for three decades. _ In general graphs, for any $k \geq 2$, we construct a $(2k-1)(1+\epsilon)$-spanner with lightness $O(n^{1/k})$ in $O(m \alpha(m,n))$ time. This result for light spanners in general weighted graphs is surprising, as it outperforms the analog one for sparse spanners.