  Seminal works on light spanners over the years provide spanners with optimal
lightness in various graph classes, such as in general graphs, Euclidean
spanners, and minor-free graphs. 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 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.
  Among various applications and implications of our framework, we highlight
here the following:
  _ In low-dimensional Euclidean spaces, we present an $O(n\log n)$-time
construction of $(1+\epsilon)$-spanners with lightness and degree both bounded
by constants in the algebraic computation tree (ACT). Our construction resolves
a major problem in the area of geometric spanners, which was open for three
decades.
  _ In general graphs, 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. This result for light spanners in general weighted graphs is surprising,
as it outperforms the analog one for sparse spanners.
