1. School of CSE, Georgia Tech
Analysis of Real World NP-Complete Graph Problem: DCJ Median Algorithm to Find Ancestor of Genome of Three
Zhaoming Yin
School of CSE, Georgia Tech

2. Foundamentals
Sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences.
...

3. Foundamentals
In 1980s Jeffrey Palmer studied evolution of plant organelles by comparing mitochondrial genomes of the cabbage and turnip, 99% similarity between genes, These surprisingly identical gene sequences differed in gene order, This study helped pave the way to analyzing genome rearrangements in molecular evolution.
Inversion:
1 2 –6 –
Transposition:
Inverted Transposition: –

4. Foundamentals
Maximal Parsimony Phylogeny is to optimize each ancestral node of an unrooted phylogeny in terms of its three or more immediate neighbours, modern or ancestral, and to iterate across the tree until convergence of the objective function (to a local optimum) at all nodes.

5. Break Point Graph 1 2 -1 2 1 2 3 4 5 6 1 -5 -2 3 -6 -4 0/+1 1/-1 2/+2
3/-2
1/-1
0/+1
2/+2
3/-2
11/-6
0/+1
1/-1
2/+2
3/-2
4/+3
5/-3
6/+4
7/-4
8/+5
9/-5
10/+6

6. MBG/0-Matching -6 -3 -2 +5 +3 +1 -4 +4 +2 -1 +6 -5

7. Subgraph/Decomposer 1 2 3 4 5 6 1 -5 -2 3 -6 -4 1 3 5 -4 6 -2-3 -2 +5 +3 +1 -4 +4 +2 -1
Subgraph +6 -5
H-crossing

8. Adequate Subgraph
Definition: In an MBG for a set of genomes G, a connected subgraph H of size m is an adequate subgraph if cmax(H) ≥ 1/2mNG; it is strongly adequate if cmax(H) >1/2mNG. (m is the size of node in the subgraph, NG is the size of genome, which is 3 for the median of three problem).
Property: A Adequate Subgraph is simple, if it does not contain another
adequate subgraph.
Lemma: A Adequate Subgraph is a decomposer.

9. Adequate Subgraph

10. Algorithm: AS1() for each v do if v[0]=v[1] or v[0]=v[2] or v[1]=v[2]
major set
for each v do
if v[0]=v[1] or v[0]=v[2] or v[1]=v[2]
these two points are AS;
the edge conncecting them is
major set;
endif
endfor

11. Adequate Subgraph √ √

12. Algorithm: AS2() c c c1 c2 c2 c1 (1) (2) c2 c1 c c1 c c2 (1) (2)
for each color c do
for each v do
if v[c1][c]=v[c][c2](1) or v[c2][c]=v[c][c1] (2) or
v[c2][c1]=v[c][c2] (3) or v[c1][c2]=v[c][c1]
v,v[c],v[c1],v[c2] are AS;
(1), major set is (v,v[c1]) and (v[c],v[c2]) or
(2), major set is (v,v[c2]) and (v[c],v[c1]) or
(3), major set is (v,v[c]) and (v[c1],v[c][c2]) or
(4), major set is (v,v[c]) and (v[c2],v[c][c1])
endif
endfor

13. Algorithm: AS2() c c2 c1 c1 c2 c2 c1 c c (1) (2) for each color c do
for each v do
if v[c1][c]=v[c][c1] and (v[c1]=v[c][c2] || v[c1]=v[c][c2) (1) or
v[c1][c2]=v[c][c1] and (v[c1]=v[c][c2] || v[c1]=v[c][c2) (2)
v,v[c],v[c1],v[c2] are AS;
(1), major set is (v,v[c1]) and (v[c],v[c2]) or
(2), major set is (v,v[c2]) and (v[c],v[c1])
endif
endfor

14. Algorithm: AS2() for each color c do for each v do
if v[c1][c]=v[c][c1] and (v[c2][c]=v[c][c2] and
v[c1]!=v[c][c2] and v[c2] !v[c][c1]
v,v[c],v[c1],v[c2] are AS;
(1), major set is (v,v[c1]) and (v[c],v[c2])
endif
endfor c2 c c1

15. Algorithm: AS2() In this case, there are two major sets
for each color c do
for each v do
if v[c1][c]=v[c][c1] and type three is not find
v,v[c],v[c1],v[c2] are AS;
(1), major set is (v,v[c1]) and (v[c],v[c2])
and (v,v[c]) and (v[c1],v[c][c1])
endif
endfor
In this case,
there are two major sets c c1

16. Adequate Subgraph √ √ √ √ √ √

17. Algorithm: AS4()--type 5-3-5c2 c1
po1
core
po2 c0
po11
po22
po0

18. Adequate Subgraph √ √ √ √ √ √ √ √ √ √

19. Algorithm: AS4()

20. Adequate Subgraph √ √ √ √ √ √ √ √ √ √ √ √ √

21. Algorithm: AS4()

22. Algorithm: AS4()

23. Adequate Subgraph √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √

24. Algorithm: Shrink() 11 5 3 8 4 7 6 2 1 10 9

25. Algorithm: Shrink() 11 5 2 5 3 4 6 1 7 6

26. Algorithm: Shrink() 11 5 2 5 3 4 6 1 7 6

27. Branch and Bound Algorithm

28. Branch and Bound Algorithm(1)
If there is no brach
that has the current
upper bound, decrease
it.
No element in the
memory, load others
from disk.

29. Branch and Bound Algorithm(2)
Get a intermediate sub-
graph, and check if it could
be trimed, or it is the final
solution.
If too much elems in the memory
store them in the disk.

30. Upperbound and Lowerbound-Upperbound
DCJ distance between genomes obey triangular inequality. So:
Given Three genomes G1 G2 G3, the median genome will have the
distance between them:
Because the distance is defined by:
therefore, the upperbound for circle number is:

31. Upperbound and Lowerbound-Upperbound
DCJ distance between genomes obey triangular inequality. So:
Given Three genomes G1 G2 G3, the median genome will have the
distance between them:
Because the distance is defined by:
therefore, the upperbound for circle number is:

32. Best First Search
Because best first search can ensure that the searching space is minimal.
However, it needs much space to store the foot print.
Which makes the branch and bound algorithm an I/O bound algorithm. 1 2 3 4 k
k+1
k+1 5 6 7 7 3 1 8 9 10 9 5 2 10 6 4 8

33. Reference
[1] Andrew Wei Xu and David Sankoff, Decompositions of multiple breakpoint graphs and rapid exact solutions to the median problem., K.A. Crandal l and J. Lagergren (Eds.): Proceedings of the Workshop on Algorithms in Bioinformatics, WABI 2008, Lecture Notes in Bioinformatics 5251,Springer.
[2] Yancopoulos, S., Attie, O., Friedberg, R.: E?cient sorting of genomic permutations by translocation, inversion and block interchange. Bioinform. 21, 3340ĺC3346 (2005)
[3] Andrew Wei Xu, A Fast and Exact Algorithm for the Median of three Problem: a Graph Decomposition Approach., Journal of computational biology, 2009, 16(10), 1-13.