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

Abstract

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):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).

Near-Optimal Spanners for General Graphs in (Nearly) Linear Time

“…We obtain the result of Theorem 1.2 by strengthening the framework of [LS21] for fast constructions of light spanners to achieve a near-optimal bound on the sparsity as well. To this end, we plug the ideas used in the proof of Theorem 1.1, in conjunction with numerous new insights, on top of the framework of [LS21] in a highly nontrivial way. Our MST-clustering approach plays a key role not just in the proof of Theorem 1.1, but also in the proof of Theorem 1.2; refer to Section 1.1 for more details.…”

“…There was a sequence of works from recent years on light spanners [ES16, ENS14, CW16, FS20, EN18, ADF + 19, LS21]. In particular, a construction of (2k − 1)(1 + )spanners with a near-optimal lightness of O(n 1/k •poly(1/ )) within a runtime of O(mα(m, n) was presented recently [LS21], where α(•, •) is the inverse-Ackermann function; on the negative side, the sparsity of the construction of [LS21] is unbounded. As mentioned, the construction of [CW16] achieves a near-optimal bound of O(n 1/k •poly(1/ )) on both sparsity and lightness, but its runtime is far from linear.…”

“…Finally, we show how to construct a spanner with near-optimal sparsity and lightness. Our construction builds on the fast construction of spanners with near-optimal lightness in [LS21]. The construction of [LS21] has a preprocessing step and a main construction step.…”

“…Our construction builds on the fast construction of spanners with near-optimal lightness in [LS21]. The construction of [LS21] has a preprocessing step and a main construction step. In the preprocessing step, every edge of weight at most w(MST) m is added to the spanner.…”

“…The main construction step is based on a cluster hierarchy. However, clusters in [LS21] are "equipped" with a potential function, and the challenge of the cluster construction is to guarantee a sufficient reduction in the potential values between two consecutive levels of the hierarchy. A cluster graph is also used to select a subset of edges in E σ i to add to the spanner.…”