{"title":"新型加重扳手","authors":"An La, Hung Le","doi":"arxiv-2408.14638","DOIUrl":null,"url":null,"abstract":"Ahmed, Bodwin, Sahneh, Kobourov, and Spence (WG 2020) introduced additive\nspanners for weighted graphs and constructed (i) a $+2W_{\\max}$ spanner with\n$O(n^{3/2})$ edges and (ii) a $+4W_{\\max}$ spanner with $\\tilde{O}(n^{7/5})$\nedges, and (iii) a $+8W_{\\max}$ spanner with $O(n^{4/3})$ edges, for any\nweighted graph with $n$ vertices. Here $W_{\\max} = \\max_{e\\in E}w(e)$ is the\nmaximum edge weight in the graph. Their results for $+2W_{\\max}$, $+4W_{\\max}$,\nand $+8W_{\\max}$ match the state-of-the-art bounds for the unweighted\ncounterparts where $W_{\\max} = 1$. They left open the question of constructing\na $+6W_{\\max}$ spanner with $O(n^{4/3})$ edges. 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. We obtain, among other things, an algorithm for\nconstructing a $+(6+\\epsilon)W_{\\max}$ spanner of\n$\\tilde{O}(\\frac{n^{4/3}}{\\epsilon})$ edges in $\\tilde{O}(n^2)$ time.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New weighted additive spanners\",\"authors\":\"An La, Hung Le\",\"doi\":\"arxiv-2408.14638\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Ahmed, Bodwin, Sahneh, Kobourov, and Spence (WG 2020) introduced additive\\nspanners for weighted graphs and constructed (i) a $+2W_{\\\\max}$ spanner with\\n$O(n^{3/2})$ edges and (ii) a $+4W_{\\\\max}$ spanner with $\\\\tilde{O}(n^{7/5})$\\nedges, and (iii) a $+8W_{\\\\max}$ spanner with $O(n^{4/3})$ edges, for any\\nweighted graph with $n$ vertices. Here $W_{\\\\max} = \\\\max_{e\\\\in E}w(e)$ is the\\nmaximum edge weight in the graph. Their results for $+2W_{\\\\max}$, $+4W_{\\\\max}$,\\nand $+8W_{\\\\max}$ match the state-of-the-art bounds for the unweighted\\ncounterparts where $W_{\\\\max} = 1$. They left open the question of constructing\\na $+6W_{\\\\max}$ spanner with $O(n^{4/3})$ edges. 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. We obtain, among other things, an algorithm for\\nconstructing a $+(6+\\\\epsilon)W_{\\\\max}$ spanner of\\n$\\\\tilde{O}(\\\\frac{n^{4/3}}{\\\\epsilon})$ edges in $\\\\tilde{O}(n^2)$ time.\",\"PeriodicalId\":501525,\"journal\":{\"name\":\"arXiv - CS - Data Structures and Algorithms\",\"volume\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Data Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.14638\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Data Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.14638","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.