Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any ddimensional n-point Euclidean space, there exists a (1 + )-spanner with n · O( −d+1 ) edges and lightness (normalized weight) O( −2d ). 1 Surprisingly, the fundamental question of whether or not these dependencies on and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus is tiny). In the most extreme case is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n.The state-of-the-art bounds n · O( −d+1 ) and O( −2d ) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is "near-optimal"; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date.The contribution of this paper is two-fold.1 Introduction Background and motivationSparse spanners. Let P be a set of n points in R d , d ≥ 2, and consider the complete weighted graph G P = (P, P 2 ) induced by P , where the weight of any edge (x, y) ∈ P 2 is the Euclidean distance |xy| between its endpoints. Let H = (P, E) be a spanning subgraph of G P , with E ⊆ P 2 , where, as in G P , the weight function is given by the Euclidean distances. For any t ≥ 1, H is called a t-spanner for P if for every x, y ∈ P , the distance d G (x, y) between x and y in G is at most t|xy|; the parameter t is called the stretch of the spanner and the most basic goal is to get it down to 1 + , for arbitrarily small > 0, 1 The lightness of a spanner is the ratio of its weight and the MST weight. without using too many edges. Euclidean spanners were introduced in the pioneering SoCG'86 paper of Chew [16], who showed that O(n) edges can be achieved with stretch √ 10, and later improved the stretch bound to 2 [17]. The first Euclidean spanners with stretch 1 + , for an arbitrarily small > 0, were presented independently in the seminal works of Clarkson [18] (FOCS'87) and Keil [38] (see also [39]), which introduced the Θ-graph in R 2 and R 3 , and soon afterwards was generalized for any R d in [45,2]. The Θ-graph is a natural variant of the Yao graph, introduced by Yao [51] in 1982, where, roughly speaking, the space R d around each point p ∈ P is partitioned into cones of angle Θ each, and then edges are added between each point p ∈ P and its closest points in each of the cones centered around it. The Θ-graph is defined similarly, where, instead of connecting p to its closest point in each cone, we connect it to a point whose orthogonal projection to some fixed ray contained in the cone is closest to p....

We show that the greedy spanner algorithm constructs a (1+ )-spanner of weight −O(d) w(MST) for a point set in metrics of doubling dimension d, resolving an open problem posed by Gottlieb [11]. Our result generalizes the result by Narasimhan and Smid [15] who showed that a point set in d-dimension Euclidean space has a (1+ )-spanner of weight at most −O(d) w(MST). Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.

We show that every H-minor-free graph has a light (1+ )-spanner, resolving an open problem of Grigni and Sissokho [13] and proving a conjecture of Grigni and Hung [12]. Our lightness bound iswhere σ H = |V (H)| log |V (H)| is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi [7] is an efficient PTAS whose running time is 2where O H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs [6].

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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).