backtracing

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’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)