Forwards[x,y] is

1. n = length(x); m = length(y)

2. Edit[Prefix[0,0]] = 0;

3. For all i from 1 to n:

 Edit[Prefix[x,i],Prefix[y,0]] = Edit[Prefix[x,i-1],Prefix[y,0]]

 + Del(x_i)

4. For all j from 1 to m:

 A. Edit[Prefix[x,0],Prefix[y,j]] = Edit[Prefix[x,0],Prefix[y,j-1]]

 + Ins(y_j)

 B. For all i from 1 to n, execute the following steps:

 i. Edit[Prefix[x,i],Prefix[y,j]] =

 min{Edit[Prefix[x,i-1],Prefix[y,j]] + Del(x_i),

 Edit[Prefix[x,i-1],Prefix[y,j-1]] + Sub(x_i,y_j),

 Edit[Prefix[x,i],Prefix[y,j-1]] + Ins(y_j)}

 ii. Erase Edit[Prefix[x,i-1],Prefix[y,j-1]]

 C. Erase Edit[Prefix[x,i-1],Prefix[y,j]]

5. RETURN Edit[x] %% an array of length m+1

Backwards[x,y] is

1. n = length(x); m = length(y)

2. For all i from 1 to n:

 Edit[Suffix[x,i],Suffix[y,0]] = 0

3. For all j from 1 to m:

 A. Edit[Suffix[x,0],Suffix[y,j]] = Edit[Suffix[x,n],Suffix[y,j-1]] +

 Ins(y_{m-j+1})

 B. For all i from 1 to n:

 i. Edit[Suffix[x,i],Suffix[y,j]] =

 min{Edit[Suffix[x,i-1],Suffix[y,j]] + Del(x_{n-i-1}),

 Edit[Suffix[x,i-1],Suffix[y,j-1]] +

 Sub(x_{n-i-1},y_{m-j+1}),

 Edit[Suffix[x,i],Suffix[y,j-1]] + Ins(y_{m-j+1})}

 ii. Erase Edit[Suffix[x,i-1],Suffix[y,j-1]]

 C. Erase Edit[Suffix[x,i-1],Suffix[y,j]]

4. RETURN Edit[x] %% an array of length m+1