Levenshtein Distance - Definition

Definition

Mathematically, the Levenshtein distance between two strings is given by where

\qquad\operatorname{lev}_{a,b}(i,j) = \begin{cases} 0 &, i=j=0 \\ i &, j = 0 \text{ and } i > 0 \\ j &, i = 0 \text{ and } j > 0 \\ \min \begin{cases} \operatorname{lev}_{a,b}(i-1,j) + 1 \\ \operatorname{lev}_{a,b}(i,j-1) + 1 \\ \operatorname{lev}_{a,b}(i-1,j-1) + \end{cases} &, \text{ else} \end{cases}

Note that the first element in the minimum corresponds to deletion(from to ), the second to insertion and the third to match or mismatch, depending on whether the respective symbols are the same.

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