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)