Fast Approximate Counting of Cycles
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866929519852519424 |
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| author | Censor-Hillel, Keren Even, Tomer Williams, Virginia Vassilevska |
| author_facet | Censor-Hillel, Keren Even, Tomer Williams, Virginia Vassilevska |
| contents | We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a $(1 \pm \eps)$-approximation in $\tilde{O}(n^ω/t^{ω-2})$ time, where $t$ is the unknown number of triangles in the given $n$ node graph and $ω<2.372$ is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an $n\times n/t$ matrix by an $n/t \times n$ matrix. We then extend our framework to obtain the first nontrivial $(1 \pm \eps)$-approximation algorithms for the number of $h$-cycles in a graph, for any constant $h\geq 3$. Our running time is
\[\tilde{O}(\mathsf{MM}(n,n/t^{1/(h-2)},n)), \textrm{the time to multiply } n\times \frac{n}{t^{1/(h-2)}} \textrm{ by } \frac{n}{t^{1/(h-2)}}\times n \textrm{ matrices}.\]
Finally, we show that under popular fine-grained hypotheses, this running time is optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast Approximate Counting of Cycles Censor-Hillel, Keren Even, Tomer Williams, Virginia Vassilevska Data Structures and Algorithms We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a $(1 \pm \eps)$-approximation in $\tilde{O}(n^ω/t^{ω-2})$ time, where $t$ is the unknown number of triangles in the given $n$ node graph and $ω<2.372$ is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an $n\times n/t$ matrix by an $n/t \times n$ matrix. We then extend our framework to obtain the first nontrivial $(1 \pm \eps)$-approximation algorithms for the number of $h$-cycles in a graph, for any constant $h\geq 3$. Our running time is \[\tilde{O}(\mathsf{MM}(n,n/t^{1/(h-2)},n)), \textrm{the time to multiply } n\times \frac{n}{t^{1/(h-2)}} \textrm{ by } \frac{n}{t^{1/(h-2)}}\times n \textrm{ matrices}.\] Finally, we show that under popular fine-grained hypotheses, this running time is optimal. |
| title | Fast Approximate Counting of Cycles |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2409.19292 |