1. Dynamic Programming: Edit Distance

2. Aligning Sequences without Insertions and Deletions: Hamming Distance
Given two sequences v and w :
v : A T
w : A T
The Hamming distance: dH(v, w) = 8 is large but the sequences are very similar

3. Aligning Sequences with Insertions and Deletions
By shifting one sequence over one position:
v : A T --
w : -- A T
The edit distance: dH(v, w) = 2.
Hamming distance neglects insertions and deletions in the sequences

4. Edit Distance
Levenshtein (1966) introduced edit distance between two strings as the minimum number of elementary operations (insertions, deletions, and substitutions) to transform one string into the other
d(v,w) = MIN number of elementary operations
to transform v  w

5. Edit Distance vs Hamming Distance
always compares
i-th letter of v with
i-th letter of w
V = ATATATAT
W = TATATATA
Hamming distance:
d(v, w)=8
Computing Hamming distance
is a trivial task.

6. Edit Distance vs Hamming Distance
may compare
i-th letter of v with
j-th letter of w
Hamming distance
always compares
i-th letter of v with
i-th letter of w
V = - ATATATAT
V = ATATATAT
W = TATATATA
W = TATATATA
Hamming distance: Edit distance:
d(v, w)= d(v, w)=2
(one insertion and one deletion)
How to find what j goes with what i ???

7. Edit Distance: Example
TGCATAT  ATCCGAT in 5 steps
TGCATAT  (delete last T)
TGCATA  (delete last A)
TGCAT  (insert A at front)
ATGCAT  (substitute C for 3rd G)
ATCCAT  (insert G before last A)
ATCCGAT (Done)

8. Alignment as a Path in the Edit Graph1 2 3 4 5 6 7 G A T C w v
A T _ G T T A T _
A T C G T _ A _ C
(0,0) , (1,1) , (2,2), (2,3), (3,4), (4,5), (5,5), (6,6), (7,6), (7,7)
- Corresponding path -

9. Alignment as a Path in the Edit Graph1 2 3 4 5 6 7 G A T C w v
Old Alignment
v= AT_GTTAT_
w= ATCGT_A_C
New Alignment
v= AT_GTTAT_
w= ATCG_TA_C

10. Dynamic programming (Cormen et al.)
Optimal substructure: The optimal solution to the problem contains within it optimal solutions to subproblems.
Overlapping subproblems: The optimal solutions to subproblems (“subsolutions”) overlap. These subsolutions are computed over and over again when computing the global optimal solution.
Optimal substructure: We compute minimum distance of substrings in order to compute the minimum distance of the entire string.
Overlapping subproblems: Need most distances of substrings 3 times (moving right, diagonally, down)

11. { Dynamic Programming si,j = si-1, j-1+ (vi != wj) min si-1, j +1

12. Levenshtein distance: Computation

13. Levenshtein distance: algorithm