Towards practical FPRAS for #NFA: Exploiting the Power of Dependence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913919089508352 |
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| author | Meel, Kuldeep S. de Colnet, Alexis |
| author_facet | Meel, Kuldeep S. de Colnet, Alexis |
| contents | #NFA refers to the problem of counting the words of length $n$ accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of $\tilde{O}(n^{17}m^{17}\varepsilon^{-14}\log(δ^{-1}))$, where $m$ is the number of states in the automaton, $δ\in (0,1]$ is the confidence parameter, and $\varepsilon > 0$ is the tolerance parameter (typically smaller than $1$). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity $\tilde{O}((n^{10}m^2 + n^6m^3)\varepsilon^{-4}\log^2(δ^{-1}))$. The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation.
In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of $O(n^2m^3\log(nm)\varepsilon^{-2}\log(δ^{-1}))$. Observe that evaluating whether a word of length $n$ is accepted by an NFA has a time complexity of $O(nm^2)$. Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards practical FPRAS for #NFA: Exploiting the Power of Dependence Meel, Kuldeep S. de Colnet, Alexis Data Structures and Algorithms #NFA refers to the problem of counting the words of length $n$ accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of $\tilde{O}(n^{17}m^{17}\varepsilon^{-14}\log(δ^{-1}))$, where $m$ is the number of states in the automaton, $δ\in (0,1]$ is the confidence parameter, and $\varepsilon > 0$ is the tolerance parameter (typically smaller than $1$). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity $\tilde{O}((n^{10}m^2 + n^6m^3)\varepsilon^{-4}\log^2(δ^{-1}))$. The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation. In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of $O(n^2m^3\log(nm)\varepsilon^{-2}\log(δ^{-1}))$. Observe that evaluating whether a word of length $n$ is accepted by an NFA has a time complexity of $O(nm^2)$. Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks. |
| title | Towards practical FPRAS for #NFA: Exploiting the Power of Dependence |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.23561 |