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 [21], Euclidean spanners [33] and minorfree graphs [12]. 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 + ). (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 (for simplicity assume > 0 is fixed): • In general graphs: A nearly linear-time algorithm for constructing light spanners. Specifically, for any k ≥ 2, we construct a (2k − 1)(1 + )-spanner with lightness O(n) in O(mα(m,n)) time, where α(·, ·) is the inverse-Ackermann function; the lightness bound is optimal assuming Erdős’ girth conjecture (up to the -dependency). Since O(mα(m,n)) = O(m + n log∗ n), the runtime is linear in m when m = Ω(n log∗ n). The previous state-of-the-art runtime of such a spanner is super-quadratic in n [21, 1]. • In low-dimensional Euclidean spaces: An O(n log n)-time construction of (1 + )-spanners with lightness and degree both bounded by constants in the basic algebraic computation tree (ACT) model. This construction is optimal (up to dependencies on and the dimension) with respect to all the involved quality measures — runtime, lightness and degree — and it resolves a major problem in the area of geometric spanners, which was open for three decades (cf. [19, 3, 50, 63]). • In unit-disk graphs in R: An O(n log n)-time construction of (1 + )-spanners with constant lightness and degree, in the ACT model. This construction too is optimal with respect to all the involved quality measures. This is the first o(n)-time spanner construction for unit disk graphs with a nontrivial lightness bound. • In minor-free graphs: A linear-time algorithm for constructing (1 + )-spanners with constant lightness. This is the first o(n)-time spanner construction for minor-free graphs (and even for some of its sub-classes, such as bounded treewidth graphs), with a nontrivial lightness bound. Remark. Our follow-up paper builds on the foundations of the theory laid in the current paper, aiming to achieve lightness bounds with optimal dependencies on the involved parameters, most notably , but also others such as the dimension (in Euclidean spaces) or the minor size (in minor-free graphs). The lightness is a normalized notion of weight: a graph’s lightness is the ratio of its weight and the MST weight. ar X iv :2 10 6. 15 59 6v 2 [ cs . C G ] 2 J ul 2 02 1