edit distance with DP

Goal: search for a path (sequence of edits) from start to final string, whereby:

• initial state is the word we are transforming
• operators: insert, delete, substitute
• goal state: the word we end up at
• path cost: cost of the path we are trying to minimize

Sequence of all edits is huge! so DP.

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:

• \(D(i,0) = i, \forall i\)
• \(D(0,j) = j, \forall j\)

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)