Computable Bounds and Monte Carlo Estimates of the Expected Edit Distance
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| Formato: | Preprint |
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2022
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| _version_ | 1866911829091942400 |
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| author | Bilardi, Gianfranco Schimd, Michele |
| author_facet | Bilardi, Gianfranco Schimd, Michele |
| contents | The edit distance is a metric of dissimilarity between strings, widely applied in computational biology, speech recognition, and machine learning. Let $e_k(n)$ denote the average edit distance between random, independent strings of $n$ characters from an alphabet of size $k$. For $k \geq 2$, it is an open problem how to efficiently compute the exact value of $α_{k}(n) = e_k(n)/n$ as well as of $α_{k} = \lim_{n \to \infty} α_{k}(n)$, a limit known to exist.
This paper shows that $α_k(n)-Q(n) \leq α_k \leq α_k(n)$, for a specific $Q(n)=Θ(\sqrt{\log n / n})$, a result which implies that $α_k$ is computable. The exact computation of $α_k(n)$ is explored, leading to an algorithm running in time $T=\mathcal{O}(n^2k\min(3^n,k^n))$, a complexity that makes it of limited practical use.
An analysis of statistical estimates is proposed, based on McDiarmid's inequality, showing how $α_k(n)$ can be evaluated with good accuracy, high confidence level, and reasonable computation time, for values of $n$ say up to a quarter million. Correspondingly, 99.9\% confidence intervals of width approximately $10^{-2}$ are obtained for $α_k$.
Combinatorial arguments on edit scripts are exploited to analytically characterize an efficiently computable lower bound $β_k^*$ to $α_k$, such that $ \lim_{k \to \infty} β_k^*=1$. In general, $β_k^* \leq α_k \leq 1-1/k$; for $k$ greater than a few dozens, computing $β_k^*$ is much faster than generating good statistical estimates with confidence intervals of width $1-1/k-β_k^*$.
The techniques developed in the paper yield improvements on most previously published numerical values as well as results for alphabet sizes and string lengths not reported before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07644 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Computable Bounds and Monte Carlo Estimates of the Expected Edit Distance Bilardi, Gianfranco Schimd, Michele Formal Languages and Automata Theory Data Structures and Algorithms Probability 68R05 (Primary) 41A25, 68W32 (Secondary) G.3; F.2.2 The edit distance is a metric of dissimilarity between strings, widely applied in computational biology, speech recognition, and machine learning. Let $e_k(n)$ denote the average edit distance between random, independent strings of $n$ characters from an alphabet of size $k$. For $k \geq 2$, it is an open problem how to efficiently compute the exact value of $α_{k}(n) = e_k(n)/n$ as well as of $α_{k} = \lim_{n \to \infty} α_{k}(n)$, a limit known to exist. This paper shows that $α_k(n)-Q(n) \leq α_k \leq α_k(n)$, for a specific $Q(n)=Θ(\sqrt{\log n / n})$, a result which implies that $α_k$ is computable. The exact computation of $α_k(n)$ is explored, leading to an algorithm running in time $T=\mathcal{O}(n^2k\min(3^n,k^n))$, a complexity that makes it of limited practical use. An analysis of statistical estimates is proposed, based on McDiarmid's inequality, showing how $α_k(n)$ can be evaluated with good accuracy, high confidence level, and reasonable computation time, for values of $n$ say up to a quarter million. Correspondingly, 99.9\% confidence intervals of width approximately $10^{-2}$ are obtained for $α_k$. Combinatorial arguments on edit scripts are exploited to analytically characterize an efficiently computable lower bound $β_k^*$ to $α_k$, such that $ \lim_{k \to \infty} β_k^*=1$. In general, $β_k^* \leq α_k \leq 1-1/k$; for $k$ greater than a few dozens, computing $β_k^*$ is much faster than generating good statistical estimates with confidence intervals of width $1-1/k-β_k^*$. The techniques developed in the paper yield improvements on most previously published numerical values as well as results for alphabet sizes and string lengths not reported before. |
| title | Computable Bounds and Monte Carlo Estimates of the Expected Edit Distance |
| topic | Formal Languages and Automata Theory Data Structures and Algorithms Probability 68R05 (Primary) 41A25, 68W32 (Secondary) G.3; F.2.2 |
| url | https://arxiv.org/abs/2211.07644 |