{"title":"Tight Bounds for Vertex Connectivity in Dynamic Streams","authors":"Sepehr Assadi, Vihan Shah","doi":"10.48550/arXiv.2211.04685","DOIUrl":null,"url":null,"abstract":"We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \\geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\\widetilde{O}(k n)$ space. Our upper bound matches the known $\\Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $\\widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a\"certificate\"of $k$-vertex-connectivity as a union of $O(k^2\\log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"50 1","pages":"213-227"},"PeriodicalIF":0.0000,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2211.04685","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\widetilde{O}(k n)$ space. Our upper bound matches the known $\Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $\widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a"certificate"of $k$-vertex-connectivity as a union of $O(k^2\log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.