Improved girth approximation in weighted undirected graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912490653220864 |
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| author | Kadria, Avi Roditty, Liam Sidford, Aaron Williams, Virginia Vassilevska Zwick, Uri |
| author_facet | Kadria, Avi Roditty, Liam Sidford, Aaron Williams, Virginia Vassilevska Zwick, Uri |
| contents | Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved girth approximation in weighted undirected graphs Kadria, Avi Roditty, Liam Sidford, Aaron Williams, Virginia Vassilevska Zwick, Uri Data Structures and Algorithms Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}. |
| title | Improved girth approximation in weighted undirected graphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.13869 |