New algorithms for girth and cycle detection
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916959611781120 |
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| author | Roditty, Liam Trabelsi, Plia |
| author_facet | Roditty, Liam Trabelsi, Plia |
| contents | Let $G=(V,E)$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Let $g$ be the girth of $G$, that is, the length of a shortest cycle in $G$. We present a randomized algorithm with a running time of $\tilde{O}\big(\ell \cdot n^{1 + \frac{1}{\ell - \varepsilon}}\big)$ that returns a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil - 2 \left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor, $ where $\ell \geq 2$ is an integer and $\varepsilon \in [0,1]$, for every graph with $g = polylog(n)$.
Our algorithm generalizes an algorithm of Kadria \etal{} [SODA'22] that computes a cycle of length at most $4\left\lceil \frac{g}{2} \right\rceil - 2\left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor $ in $\tilde{O}\big(n^{1 + \frac{1}{2 - \varepsilon}}\big)$ time. Kadria \etal{} presented also an algorithm that finds a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil $ in $\tilde{O}\big(n^{1 + \frac{1}{\ell}}\big)$ time, where $\ell$ must be an integer. Our algorithm generalizes this algorithm, as well, by replacing the integer parameter $\ell$ in the running time exponent with a real-valued parameter $\ell - \varepsilon$, thereby offering greater flexibility in parameter selection and enabling a broader spectrum of combinations between running times and cycle lengths.
We also show that for sparse graphs a better tradeoff is possible, by presenting an $\tilde{O}(\ell\cdot m^{1+1/(\ell-\varepsilon)})$ time randomized algorithm that returns a cycle of length at most $2\ell(\lfloor \frac{g-1}{2}\rfloor) - 2(\lfloor \varepsilon \lfloor \frac{g-1}{2}\rfloor \rfloor+1)$, where $\ell\geq 3$ is an integer and $\varepsilon\in [0,1)$, for every graph with $g=polylog(n)$.
To obtain our algorithms we develop several techniques and introduce a formal definition of hybrid cycle detection algorithms. [...] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New algorithms for girth and cycle detection Roditty, Liam Trabelsi, Plia Data Structures and Algorithms Let $G=(V,E)$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Let $g$ be the girth of $G$, that is, the length of a shortest cycle in $G$. We present a randomized algorithm with a running time of $\tilde{O}\big(\ell \cdot n^{1 + \frac{1}{\ell - \varepsilon}}\big)$ that returns a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil - 2 \left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor, $ where $\ell \geq 2$ is an integer and $\varepsilon \in [0,1]$, for every graph with $g = polylog(n)$. Our algorithm generalizes an algorithm of Kadria \etal{} [SODA'22] that computes a cycle of length at most $4\left\lceil \frac{g}{2} \right\rceil - 2\left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor $ in $\tilde{O}\big(n^{1 + \frac{1}{2 - \varepsilon}}\big)$ time. Kadria \etal{} presented also an algorithm that finds a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil $ in $\tilde{O}\big(n^{1 + \frac{1}{\ell}}\big)$ time, where $\ell$ must be an integer. Our algorithm generalizes this algorithm, as well, by replacing the integer parameter $\ell$ in the running time exponent with a real-valued parameter $\ell - \varepsilon$, thereby offering greater flexibility in parameter selection and enabling a broader spectrum of combinations between running times and cycle lengths. We also show that for sparse graphs a better tradeoff is possible, by presenting an $\tilde{O}(\ell\cdot m^{1+1/(\ell-\varepsilon)})$ time randomized algorithm that returns a cycle of length at most $2\ell(\lfloor \frac{g-1}{2}\rfloor) - 2(\lfloor \varepsilon \lfloor \frac{g-1}{2}\rfloor \rfloor+1)$, where $\ell\geq 3$ is an integer and $\varepsilon\in [0,1)$, for every graph with $g=polylog(n)$. To obtain our algorithms we develop several techniques and introduce a formal definition of hybrid cycle detection algorithms. [...] |
| title | New algorithms for girth and cycle detection |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.02061 |