An \emph{additive $+\beta$ spanner} of a graph $G$ is a subgraph which preserves shortest paths up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al.\ 2019 and 2020, Ahmed et al.\ 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al.\ 2020] provided constructions of sparse spanners with \emph{global} error $\beta = cW$, where $W$ is the maximum edge weight in $G$ and $c$ is constant. We improve these to \emph{local} error by giving spanners with additive error $+cW(s,t)$ for each vertex pair $(s,t)$, where $W(s, t)$ is the maximum edge weight along the shortest $s$--$t$ path in $G$. These include pairwise $+(2+\eps)W(\cdot,\cdot)$ and $+(6+\eps) W(\cdot, \cdot)$ spanners over vertex pairs $\Pc \subseteq V \times V$ on $O_{\eps}(n|\Pc|^{1/3})$ and $O_{\eps}(n|\Pc|^{1/4})$ edges for all $\eps > 0$, which extend previously known unweighted results up to $\eps$ dependence, as well as an all-pairs $+4W(\cdot,\cdot)$ spanner on $O(n^{7/5})$ edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its \emph{lightness}, defined as the total edge weight of the spanner divided by the weight of an MST of $G$. We provide a $+\eps W(\cdot,\cdot)$ spanner with $O_{\eps}(n)$ lightness, and a $+(4+\eps) W(\cdot,\cdot)$ spanner with $O_{\eps}(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

中文翻译：

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具有局部加性误差的加权稀疏和轻量扳手

图$ G $的\ emph {additive $ + \ beta $ spanner}是一个子图，它保留了最短路径，直到$ + \ beta $误差为止。在非加权图中对加法扳手进行了很好的研究，但直到最近才在加权图中得到关注[Elkin等，2019和2020，Ahmed等，2020]。本文对加权加力扳手理论做出了两个新的贡献。对于加权图，[Ahmed et al。\ 2020]提供了具有\ emph {global}错误$ \ beta = cW $的稀疏扳手构造，其中$ W $是$ G $中的最大边权重，$ c $是恒定的。通过为扳手提供每个顶点对$（s，t）$附加误差$ + cW（s，t）$，我们将这些误差提高到\ emph {local}错误，其中$ W（s，t）$是最大边沿$ G $中最短的$ s $-$ t $路径的权重。这些包括成对的$ +（2+ \ eps）W（\ cdot，\ cdot）$和$ +（6+ \ eps）W（\ cdot，\ cdot）$顶点对上的扳手$ \ Pc \ subseteq V \ $ O _ {\ eps}（n | \ Pc | ^ {1/3}）$和$ O _ {\ eps}（n | \所有\\ eps> 0 $的Pc | ^ {1/4}）$边，将先前已知的未加权结果扩展到$ \ eps $依赖性，以及全对$ + 4W（\ cdot，\ cdot ）$扳手位于$ O（n ^ {7/5}）$边上。除了稀疏性之外，在加权图中测量扳手质量的另一种自然方法是通过\ emph {lightness}定义为扳手的总边缘权重除以MST的权重$ G $。我们提供$ + \ eps W（\ cdot，\ cdot）$扳手和$ O _ {\ eps}（n）$亮度，以及$ +（4+ \ eps）W（\ cdot，\ cdot）$扳手亮度为$ O _ {\ eps}（n ^ {2/3}）$。这些是第一个已知的具有非凡轻度保证的添加剂扳手。以上所有扳手都可以在多项式时间内构造。