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Given two strings $x$ and $y$ over the alphabet $\Sigma$ one defines the edit-distance $\text{ed}(x,y)$ as the minimum number of substitutions, insertions and deletions of characters required to transform $x$ into $y$.

If $m = \max ( \text{len}(x),\text{len}(y) )$ then padding with appropriate zeroes on the smaller string one can define $x$ and $y$ as both being elements of $\Sigma^m$. Now both $x$ and $y$ are vertices of the cube $\Sigma^m$.

Now isn't $\text{ed}(x,y)$ the shortest distance between $x$ and $y$ in this cube graph? Why isn't edit distance found by finding shortest paths in this cube?

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Consider the two strings 01010101 and 10101010. They have distance 8 on the cube but their edit distance is 2.

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  • $\begingroup$ Why is the edit distance 2 here? $\endgroup$ – user6818 May 17 '15 at 20:43
  • $\begingroup$ Because you allow insertions and deletions. $\endgroup$ – André Souza Lemos May 17 '15 at 20:44
  • $\begingroup$ Oh okay - I understand - you want to insert a 1 at the beginning and delete the last 1. $\endgroup$ – user6818 May 17 '15 at 20:48
  • $\begingroup$ So is there any mathematical notion of distance which corresponds to this notion of "edit distance" ? $\endgroup$ – user6818 May 17 '15 at 20:48
  • $\begingroup$ That is a new question... or a different version of your question. $\endgroup$ – André Souza Lemos May 17 '15 at 20:49

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