新型加重扳手

An La, Hung Le
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Elkin, Gitlitz, and Neiman\n(DISC 2021) made significant progress on this problem by showing that there\nexists a $+(6+\\epsilon)W_{\\max}$ spanner with $O(n^{4/3}/\\epsilon)$ edges for\nany fixed constant $\\epsilon > 0$. Indeed, their result is stronger as the\nadditive stretch is local: the stretch for any pair $u,v$ is\n$+(6+\\epsilon)W_{uv}$ where $W_{uv}$ is the maximum weight edge on the shortest\npath from $u$ to $v$. In this work, we resolve the problem posted by Ahmed et al. (WG 2020) up to a\npoly-logarithmic factor in the number of edges: We construct a $+6W_{\\max}$\nspanner with $\\tilde{O}(n^{4/3})$ edges. We extend the construction for\n$+6$-spanners of Woodruff (ICALP 2010), and our main contribution is an\nanalysis tailoring to the weighted setting. The stretch of our spanner could\nalso be made local, in the sense of Elkin, Gitlitz, and Neiman (DISC 2021). We\nalso study the fast constructions of additive spanners with $+6W_{\\max}$ and\n$+4W_{\\max}$ stretches. 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引用次数: 0

摘要

Ahmed、Bodwin、Sahneh、Kobourov 和 Spence(WG 2020)介绍了加权图的加法生成器,并构建了:(i) 具有 $O(n^{3/2})$ 条边的 $+2W_{\max}$ 生成器;(ii) 具有 $/tilde{O}(n^{7/5})$条边的 $+4W_{\max}$ 生成器;以及 (iii) 具有 $O(n^{4/3})$ 条边的 $+8W_{\max}$ 生成器、和 (iii) 对于任何具有 $n$ 个顶点的加权图,具有 $O(n^{4/3})$ 条边的 $+8W_{max}$ 抹角器。这里的 $W_{\max} = \max_{e\in E}w(e)$ 是图中的最大边重。他们对 $+2W_{\max}$、$+4W_{\max}$ 和 $+8W_{max}$ 的研究结果与 $W_{\max} = 1$ 的非加权部分的最新界限相吻合。他们对如何构建一个具有 $O(n^{4/3})$ 条边的 $+6W_{\max}$ 抹角器持开放态度。Elkin、Gitlitz和Neiman(DISC 2021)在这个问题上取得了重大进展,他们证明了对于任何固定常数$\epsilon > 0$,都存在一个具有$O(n^{4/3}/\epsilon)$边的$+(6+\epsilon)W_{max}$扳手。事实上,他们的结果更强,因为加法拉伸是局部的:任意一对 $u,v$ 的拉伸为 $+(6+\epsilon)W_{uv}$,其中 $W_{uv}$ 是 $u$ 到 $v$ 最短路径上的最大权重边。在这项工作中,我们解决了艾哈迈德等人(WG 2020)提出的问题,达到了边数的对数因子:我们用 $\tilde{O}(n^{4/3})$ 边构建了一个 $+6W_{max}$spanner 。我们扩展了伍德拉夫(Woodruff,ICALP 2010)的 $+6$spanner 的构造,我们的主要贡献是针对加权设置的分析。根据 Elkin、Gitlitz 和 Neiman(DISC 2021)的观点,我们的施展器的伸展也可以是局部的。我们还研究了具有$+6W_{\max}$和$+4W_{\max}$拉伸的加法算子的快速构造。除其他外,我们还获得了一种算法,可以在 $\tilde{O}(n^2)$ 时间内构造出 $+(6+\epsilon)W_{max}$ 的 $+(6+\epsilon)W_{max}$ 加边器。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
New weighted additive spanners
Ahmed, Bodwin, Sahneh, Kobourov, and Spence (WG 2020) introduced additive spanners for weighted graphs and constructed (i) a $+2W_{\max}$ spanner with $O(n^{3/2})$ edges and (ii) a $+4W_{\max}$ spanner with $\tilde{O}(n^{7/5})$ edges, and (iii) a $+8W_{\max}$ spanner with $O(n^{4/3})$ edges, for any weighted graph with $n$ vertices. Here $W_{\max} = \max_{e\in E}w(e)$ is the maximum edge weight in the graph. Their results for $+2W_{\max}$, $+4W_{\max}$, and $+8W_{\max}$ match the state-of-the-art bounds for the unweighted counterparts where $W_{\max} = 1$. They left open the question of constructing a $+6W_{\max}$ spanner with $O(n^{4/3})$ edges. Elkin, Gitlitz, and Neiman (DISC 2021) made significant progress on this problem by showing that there exists a $+(6+\epsilon)W_{\max}$ spanner with $O(n^{4/3}/\epsilon)$ edges for any fixed constant $\epsilon > 0$. Indeed, their result is stronger as the additive stretch is local: the stretch for any pair $u,v$ is $+(6+\epsilon)W_{uv}$ where $W_{uv}$ is the maximum weight edge on the shortest path from $u$ to $v$. In this work, we resolve the problem posted by Ahmed et al. (WG 2020) up to a poly-logarithmic factor in the number of edges: We construct a $+6W_{\max}$ spanner with $\tilde{O}(n^{4/3})$ edges. We extend the construction for $+6$-spanners of Woodruff (ICALP 2010), and our main contribution is an analysis tailoring to the weighted setting. The stretch of our spanner could also be made local, in the sense of Elkin, Gitlitz, and Neiman (DISC 2021). We also study the fast constructions of additive spanners with $+6W_{\max}$ and $+4W_{\max}$ stretches. We obtain, among other things, an algorithm for constructing a $+(6+\epsilon)W_{\max}$ spanner of $\tilde{O}(\frac{n^{4/3}}{\epsilon})$ edges in $\tilde{O}(n^2)$ time.
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