A Unified Framework of Light Spanners II: Fine-Grained Optimality

Abstract

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, 1 such as in general graphs [14], Euclidean spanners [24] and minor-free graphs [10]. 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 second of the two -it builds on the basis of the unified framework laid in the first paper, and then strengthens it to achieve more refined optimality bounds for several graph classes, i.e., the bounds remain optimal when taking into account a wider range of involved parameters, most notably , but also others such as the dimension (in Euclidean spaces) or the minor size (in minor-free graphs). Our new constructions are significantly better than the state-of-the-art for every examined graph class. Among various applications and implications of our framework, we highlight the following:For K r -minor-free graphs, we provide a (1 + )-spanner with lightness Õr, ( r + 12 ), where Õr, suppresses polylog factors of 1/ and r, improving the lightness bound Õr, ( r 3 ) of Borradaile, Le and Wulff-Nilsen [10]. We complement our upper bound with a highly nontrivial lower bound construction, for which any (1 + )-spanner must have lightness Ω( r + 12 ). Interestingly, our lower bound is realized by a geometric graph in R 2 . We note that the quadratic dependency on 1/ we proved here is surprising, as the prior work suggested that the dependency on should be around 1/ . Indeed, for minor-free graphs there is a known upper bound of lightness O(log(n)/ ), whereas subclasses of minorfree graphs, primarily graphs of genus bounded by g, are long known to admit spanners of lightness O(g/ ).