1. Dynamic Programming Computation of Edit Distance

2. Definition of Edit Distance
Edit Distance DE (X,Y) measures how close string X is to string Y.
DE(X,Y) is the cost of the minimum cost transformation t : X t Y where t is a sequence of operations (insertion, equal substitution, unequal substitution, and deletion). The cost of t is the sum of the operation costs where each operation costs 1 except for equal substitution which costs 0. A B C
The cost of this transformation is 3 which happens to be minimal.

3. Decomposition of Problem
Decomposition : Last Operation Delete, Substitute, or Insert
Atomic Problems : X prefix or Y prefix empty
Table :
Rows for 0 .. M for X prefix characters,
Columns 0 .. N for Y prefix characters
Table Entry : DE (Xi , Yj)
Composition :  = cost(Substitution) = 1 if xi != yj and 0 otherwise.
DE (Xi ,Yj ) = min{ DE (Xi-1 ,Yj ) + 1,
DE (Xl-1 ,Yj-1 ) + ,
DE (Xi ,Yj-1 ) + 1 }

4. Atomic Problems   Yi requires i insertions at a cost of I
Empty string transformed into a prefix of Y
Xi   requires i deletions at a cost of I
A prefix of X transformed into the empty string

5. Computation of DE( ababaac, bababbc )