This presentation is about how to find out the longest common subsequence of two Array.

1. Dipta Das
LONGEST COMMON
SUBSEQUENCE

2. Dipta Das
SUBSEQUENCE
A subsequence of given sequence is just the given sequence with zero or
more elements left out. Example..

3. Dipta Das
COMMON SUBSEQUENCE
Assume Two Sequence
Sequence Z is a common subsequence of X and Y if Z is a subsequence of both X and Y
Z={ B, C, A} length-3
Z={ B, C, A, B} length-4
Z={ B, C, B} length-3

4. Dipta Das
LONGEST COMMON SUBSEQUENCE
Theorem
Assume:
X=(X1, X2, X3…………………Xn)
Y=(Y1, Y2, Y3…………………Yn)
Any LCS of X and Y is Z, Z=(Z1, Z2,Z3……………..Zn)
IF Then
Xi = Yj Zk=Xi=Yj implies Zk-1 LCS of Xi-1 and Yj – 1
Xi ≠ Yj Zk ≠ Xi implies Zk-1 LCS of Xi-1 and Y
Xi ≠ Yj Zk ≠ Yj implies Zk-1 LCS of Yj-1 and X

5. Dipta Das
To Compare DNA of two (or more ) Different organisms

6. Dipta Das
EXAMPLE
Assume two DNA sequence
X = {ATGCTTC}
Y = {GCTCA}

7. Dipta Das
LCS EXAMPLE X = {ATGCTTC}
Y = {GCTCA}
A T G C T T C
G
C
T
C
A
1 2 3 4 5 6 7
1
2
3
4
5
Yj
Xi
0
0

8. Dipta Das
LCS EXAMPLE
A T G C T T C
0 0 0 0 0 0 0 0
G 0
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
1 2 3 4 5 6 7
1
2
3
4
5
Yj
Xi
0
0
Z[j,i]
Here I = 1, j = 1
Z[1,1]

9. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCAGA}
Yj
Xi
X Y
A G
Not Match
Maximum of
two box
z[J-1, i] and
[J, i-1]
1 2 3 4 5 6 7
1
2
3
4
5
0
0
Z[1,1]
Z[j-1, i]=Z[1-1, 1]= Z[0,1]
Z[j, i-1]=Z[1, 1-1]= Z[1,0]

10. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y
A G
Not Match
Lets Take from Upper one
Arrow indicate from
where you Take the
maximum.
1 2 3 4 5 6 7
1
2
3
4
5
0
0

11. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCAGA}
Yj
Xi
X Y Max
T G 0
Not Match
Lets Take from left one
Arrow indicate from
where you Take the
maximum.
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

12. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCAGA}
Yj
Xi
X Y Max
G G
Match
arrow
When match arrow will
be diagonal because we
will increment the
value of this cell
Z[i-1, j-1] + 10 = 1
1 2 3 4 5 6 7
1
2
3
4
5
0
0

13. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
G G
Match
arrow
Incremented value X[i-1] Y[j-1]
1 2 3 4 5 6 7
1
2
3
4
5
0
0
Z[I,j] = Z[3,1]

14. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {ACTCAGA}
Yj
Xi
X Y Max
C G 1
Not Match
Lets Take from left one
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

15. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
T G 1
Not Match
Lets Take from left one
arrow
0
0
1 2 3 4 5 6 7
1
2
3
4
5

16. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
T G 1
Not Match
Lets Take from left one
arrow
0
0
1 2 3 4 5 6 7
1
2
3
4
5

17. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
C G 1
Not Match
Lets Take from left one
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

18. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
A C 0
Not Match
Lets Take from left one
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

19. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
A C 0
Not Match
Lets Take from Upper one
arrow
0
0

20. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
G C 1
Not Match
Lets Take from left one
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

21. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
C C
Match
arrow
Increment Z[i-1,j-1]
1 2 3 4 5 6 7
1
2
3
4
5
0
0

22. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2
T 0
C 0
A 0
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
T C 2
Not Match
Lets Take from left one
arrow
1 2 3 4 5 6 7
1
2
3
4
5
0
0

23. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
X Y Max
T G 1
Not Match
Lets Take from left one
arrow
In the same way…
1 2 3 4 5 6 7
1
2
3
4
5
0
0

24. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
Firstly have to point out
highest value
For left and upper arrow
we will follow the
direction
For diagonal arrow we
will point out the
character for this cell.
1 2 3 4 5 6 7
1
2
3
4
5

25. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
LCS Z= G
1 2 3 4 5 6 7
1
2
3
4
5

26. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
LCS Z= GC
1 2 3 4 5 6 7
1
2
3
4
5
0
0

27. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
LCS Z= GCT
1 2 3 4 5 6 7
1
2
3
4
5

28. Dipta Das
LCS EXAMPLE
Xi A T G C T T C
YJ 0 0 0 0 0 0 0 0
G 0 0 0 1 1 1 1 1
C 0 0 0 1 2 2 2 2
T 0 0 1 1 2 3 3 3
C 0 0 1 1 2 3 3 4
A 0 1 1 1 2 3 3 4
X = {ATGCTTC}
Y = {GCTCA}
Yj
Xi
LCS Z= {GCTC}
1 2 3 4 5 6 7
1
2
3
4
5
0
0

29. Dipta Das
ANY QUESTION?

30. Dipta Das
Thank You!