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 as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). 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 (weights are normalized so that the minimum edge weight is 1). In addition, as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). showed a randomized 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 \(O(n\cdot \sqrt{\vert S\vert })\), improving the \(+4W_{max}\) result of as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). (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. It was proved by Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) that such spanners must suffer polynomially large stretches. 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 Dor D et al. (SIAM J Comput 29:1740–1759, 2000) 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.

图生成器和仿真器是稀疏结构，可以大致保留原始图的距离。虽然在加法扳手方面有大量的工作，但到目前为止，对加权图的关注很少。直到最近才由 Ahmed 等人报道。（在：Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 第 46 届国际研讨会，WG 2020，英国利兹）。为大小为\(O(n^{3/2})\)的未加权图扩展了经典的 +2（分别为 +4）扳手（分别为\(O(n^{7/5})\)）到加权设置，其中附加误差为\(+2W\)（分别为\(+4W\)）。这意味着对于每一对u ,  v，附加拉伸最多为\(+2W_{u,v}\)，其中\(W_{u,v}\)是最短\(uv\)路径上的最大边权重（权重被归一化，使得最小边权重为 1） . 此外，正如 Ahmed 等人所报道的那样。（在：Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 第 46 届国际研讨会，WG 2020，英国利兹）。展示了一个随机算法产生一个大小为\(O(n^{4/3})\)的\(+8W_{max}\)扳手，这里\(W_{max}\)是整个边的最大边权重图形。在这项工作中，我们通过为大小为\(O(n^{4/3})\)的加权图的\(+(6+\varepsilon )W\)扳手设计一个简单的确定性算法来改进后一个结果（对于任何常数\(\varepsilon >0\)），因此几乎与未加权图的大小为\(O(n^{4/3})\)的经典 +6 扳手匹配。此外，我们展示了一个大小为\(O(n\cdot \sqrt{\vert S\vert })\)的\(+(2+\varepsilon )W\)子集扳手，改进了\(+4W_{max} \)结果由 Ahmed 等人报告。（在：Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 第 46 届国际研讨会，WG 2020，英国利兹）。（具有相同的大小）。我们还展示了一个简单的随机算法，用于大小为\({\tilde{O}}(n^{4/3})\)的\(+4W\)模拟器. 此外，我们表明我们的技术适用于非常稀疏的加法扳手，它们具有线性大小。Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) 证明，此类扳手必须经受多项式大拉伸。对于加权图，我们使用简单确定性算法的变体，该算法产生线性大小\(+{\tilde{O}}(\sqrt{n}\cdot W)\)扳手，我们还获得了大小之间的权衡和伸展。最后，概括 Dor D 等人的技术。(SIAM J Comput 29:1740–1759, 2000) 对于未加权图，我们设计了一种有效的随机算法，为大小为\ ({\tilde{O}}(n^{3 /2})\)在\({\tilde{O}}(n^2)\)时间。