For instance, in spellcheck, you are more likely to confuse say \(a\) and \(e\) than \(a\) and \(b\). Therefore, sometimes we want to weight our edit distance with DP to account for these “common” paths to make certain corrections more “jarring”.

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)
```