Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently [ABSKS20] extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $O(n^{3/2})$ (resp., $O(n^{7/5})$) to the weighted setting, where the additive error is $+2W$ (resp., $+4W$). This means that for every pair $u,v$, the additive stretch is at most $+2W_{u,v}$, where $W_{u,v}$ is the maximal edge weight on the shortest $u-v$ path. In addition, [ABSKS20] showed an algorithm yielding a $+8W_{max}$ spanner of size $O(n^{4/3})$, here $W_{max}$ is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a $+(6+\varepsilon)W$ spanner for weighted graphs with size $O(n^{4/3})$ (for any constant $\varepsilon>0$), thus nearly matching the classical +6 spanner of size $O(n^{4/3})$ for unweighted graphs. Furthermore, we show a $+(2+\varepsilon)W$ subsetwise spanner of size $O(n\cdot\sqrt{|S|})$, improving the $+4W_{max}$ result of [ABSKS20] (that had the same size). We also show a simple randomized algorithm for a $+4W$ emulator of size $\tilde{O}(n^{4/3})$. In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size $+\tilde{O}(\sqrt{n}\cdot W)$ spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of [DHZ00] for unweighted graphs, we devise an efficient randomized algorithm producing a $+2W$ spanner for weighted graphs of size $\tilde{O}(n^{3/2})$ in $\tilde{O}(n^2)$ time.

翻译：平面图和模拟器都是稀疏的结构, 大约保持原始图的距离 。 虽然在添加量的比值上做了大量工作, 但目前很少注意加权图。 只有最近[ ABSKS20] 扩大了经典+2 (分别, +4) 的比值, 大小为O( n=3/2}) (重, $( =20) 美元), 到加权设置, 添加值的比值是 $+2W美元( 美元, 美元) 。 虽然在添加量上做了大量的工作, 但对于每对调量的比值值来说, 美元, 3美元, 3美元。 美元, 美元是最短的美元路径上的最大比值。 此外, [ABSKS20] 显示一个算法, 美元比值是 $8W=%。