A \emph{spanner} of a graph $G$ is a subgraph $H$ that approximately preserves shortest path distances in $G$. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured \emph{multiplicatively}. In this work, we investigate whether one can similarly extend constructions of spanners with purely \emph{additive} error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic $+2$ and $+4$ unweighted spanners (both all-pairs and pairwise) to $+2W$ and $+4W$ weighted spanners, where $W$ is the maximum edge weight. Specifically, we show that a weighted graph $G$ contains all-pairs (pairwise) $+2W$ and $+4W$ weighted spanners of size $O(n^{3/2})$ and $O(n^{7/5})$ ($O(np^{1/3})$ and $O(np^{2/7})$) respectively. For a technical reason, the $+6$ unweighted spanner becomes a $+8W$ weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that $G$ contains all-pairs (pairwise) $+8W$ weighted spanners of size $O(n^{4/3})$ ($O(np^{1/4})$).

中文翻译：

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加权添加剂扳手

图 $G$ 的 \emph{spanner} 是一个子图 $H$，它大约保留了 $G$ 中的最短路径距离。Spanner 通常用于压缩与加权输入图对应的度量空间上的计算。经典扳手结构可以无缝地处理边缘权重，只要测量误差\emph {乘法}。在这项工作中，我们研究是否可以类似地将具有纯 \emph {additive} 错误的扳手结构扩展到加权图。这些扩展不是直接的，因为关于最短路径邻域大小的关键引理对加权图失败。尽管如此，我们恢复了一个合适的摊销版本，这让我们可以证明经典的 $+2$ 和 $+4$ 未加权扳手（全对和成对）直接扩展到 $+2W$ 和 $+4W$ 加权扳手，其中 $W$ 是最大边缘权重。具体来说，我们展示了一个加权图 $G$ 包含大小为 $O(n^{3/2})$ 和 $O(n^{ 7/5})$ ($O(np^{1/3})$ 和 $O(np^{2/7})$) 分别。由于技术原因，$+6$未加权扳手变成了$+8W$加权扳手；缩小这个错误差距是一个有趣的遗留问题。也就是说，我们证明 $G$ 包含大小为 $O(n^{4/3})$ ($O(np^{1/4})$) 的所有对（成对）$+8W$ 加权扳手.