Computable Bounds and Monte Carlo Estimates of the Expected Edit Distance

TL;DR

This paper develops methods to compute and estimate the expected edit distance between random strings, providing bounds, algorithms, and statistical techniques to evaluate it efficiently for large string lengths and alphabet sizes.

Contribution

It introduces new bounds, algorithms, and statistical estimation methods for the expected edit distance, improving accuracy and efficiency over previous approaches.

Findings

Bounds on the limit of normalized expected edit distance are established.

A computationally intensive algorithm for exact values is presented.

Statistical estimates with high confidence are feasible for large string lengths.

Abstract

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 $\alpha_{k}(n) = e_k(n)/n$ as well as of $\alpha_{k} = \lim_{n \to \infty} \alpha_{k}(n)$, a limit known to exist. This paper shows that $\alpha_k(n)-Q(n) \leq \alpha_k \leq \alpha_k(n)$, for a specific $Q(n)=\Theta(\sqrt{\log n / n})$, a result which implies that $\alpha_k$ is computable. The exact computation of $\alpha_k(n)$ 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 $\alpha_k(n)$ 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 $\alpha_k$. Combinatorial arguments on edit scripts are exploited to analytically characterize an efficiently computable lower bound $\beta_k^*$ to $\alpha_k$, such that $ \lim_{k \to \infty} \beta_k^*=1$. In general, $\beta_k^* \leq \alpha_k \leq 1-1/k$; for $k$ greater than a few dozens, computing $\beta_k^*$ is much faster than generating good statistical estimates with confidence intervals of width $1-1/k-\beta_k^*$. 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.