Suggest edit — backtracing

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title: "backtracing"
source: https://www.jemoka.com/posts/kbhbacktracing/
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we need to keep two sequences aligned; so in addition to minimum edit distance we need to know how to transform one sequence into another.
To do this, we keep a pointer of what cell we came from.
This is similar to edit distance with DP, but we keep a pointer of each cell of the action: point DOWN (less j) if inserting, point LEFT (less i) if deleting, and point diagonally if substituting.
For two strings, let&rsquo;s define:
\(X\) of length \(n\) \(Y\) of length \(m\) we define some \(D(i,j)\) as the edit distance between substring \(X[1:i]\) and \(Y[1:j]\).
Let:
\(D(i,0) = i\) \(D(0,j) = j\) for i in range(1,M): for j in range(1,N): # deletion: ignoring one char of previous string d1 = D(i-1,j) + 1 # (cost) # insertion: insertion into string before using rest of j d2 = D(i,j-1) + 1 # (cost) # keep same if char is same or substitute current d3 = D(i-1,j-1) + (0 if X[i] == Y[j] else 2) # cache D(i,j) = min(d1, d2, d3)