A UNIFIED FRAMEWORK OF LIGHT SPANNERS I: FAST (YET OPTIMAL) CONSTRUCTIONS

We present a unified framework for constructing 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. Our new constructions are significantly faster than the state-of-the-art for every graph class studied in this paper; the running times of our constructions are near-linear and usually optimal. Among various applications and implications of our framework, we highlight here the following (for simplicity assume \epsilon > 0 is fixed): (1) In low-dimensional Euclidean spaces, we present a construction of (1+\epsilon)-spanners for n-point sets with lightness and degree both bounded by constants, running in O(n log n) time in the algebraic computation tree (ACT) (or real-RAM) model, which is the basic model used in computational geometry. Our construction is optimal with respect to all the involved quality measures-running time, lightness, and degree-and it resolves a major problem in the area of geometric spanners, which was open for three decades. (2) In general graphs, we present a near-linear time algorithm for constructing light spanners of graphs with n vertices and m edges. Specifically, 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, where \alpha(\cdot, \cdot) is the inverse Ackermann function; the lightness bound matches Erd\Hos' girth conjecture up to the \epsilon-dependency. (Our companion paper builds on the basis laid in this paper, aiming to achieve optimality in a more refined sense, which takes into account a wider range of involved parameters, most notably \epsilon, but also others such as the Euclidean dimension or the minor size (in minor-free graphs).)