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 a given size k. An open problem is the exact value of \(\alpha _{k}(n)= e_k(n)/n\). While it is known that, for increasing n, \(\alpha _{k}(n)\) approaches a limit \(\alpha _{k}\), the exact value of this limit is unknown, for any \(k\ge 2\). This paper presents an upper bound to \(\alpha _{k}\) based on the exact computation of some \(\alpha _k(n)\) and a lower bound to \(\alpha _{k}\) based on combinatorial arguments on edit scripts. Statistical estimates of \(\alpha _{k}(n)\) are also obtained, with analysis of error and of confidence intervals. The techniques are applied to several alphabet sizes k. In particular, for a binary alphabet, the rigorous bounds are \(0.1742 \le \alpha _2 \le 0.3693\) while the obtained estimate is \(\alpha _2 \approx 0.2888\); for a quaternary alphabet, \(0.3598 \le \alpha _4 \le 0.6318\) and \(\alpha _4 \approx 0.5180\). These values are more accurate than those previously published.
This work was partially supported by University of Padova projects CPDA152255/15 and CPGA3/13; by MIUR, the Italian Ministry of Education, University and Research, under Grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data; and by an IBM SUR Grant.
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Notes
- 1.
A similar algorithm computes the length of the LCS. The recurrence (3) becomes \(M_{i,0} = 0\), \(M_{0,j} = 0\), and \(M_{i,j} = \max {\{ M_{i-1,j-1} + (1-\xi _{i,j}) ; M_{i-1,j} ; M_{i,j-1} \}}.\)
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Schimd, M., Bilardi, G. (2019). Bounds and Estimates on the Average Edit Distance. In: Brisaboa, N., Puglisi, S. (eds) String Processing and Information Retrieval. SPIRE 2019. Lecture Notes in Computer Science(), vol 11811. Springer, Cham. https://doi.org/10.1007/978-3-030-32686-9_7
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