Dynamic Programming: Minimum Edit Distance

with 3 edits (delete the C, delete the last A and insert a C). This is the best that can be done so the minimum edit distance is 3. The MED problem is an important problem that has many applications. For example in version control systems such as git or svn when you update a file and commit it, the system does not store the new version but instead only stores the “difference” from the previous version1. This is important since often the user is only making small changes and it would be wasteful to store the whole file. Variants of the minimum edit distance problem are use to find this “difference”. Edit distance can also be used to reduce communication costs by only communicating the differences from a previous version. It turns out that edit-distance is also closely related to approximate matching of genome sequences. One might consider a greedy method that scans the sequence finding the first difference, fixing it and then moving on. Unfortunately no simple greedy method is known to work. The problem is that there can be multiple ways to fix the error—we can either delete the offending character, or insert a new one. In the example above when we get to the C in S we could either delete C or insert an A. If we greedily pick the wrong way to fix it, we might not end up with an optimal solution. Again in the example, if you inserted an A, then more than two more edits will be required. However, considering the greedy solution gives a good hint of how to find a correct solution. In particular when we get to the C in our example there were exactly two possible ways to fix it—deleting C or inserting A. This leads to the following algorithm.