The Breakpoint Graph

1
The Breakpoint Graph
1 5- 2-
4 3
2
The Breakpoint Graph
6 1 5- 2-
4 3 0

• Augment with 0 n1

3
The Breakpoint Graph
11 2 1 9 10 3 4 8
7 6 5 0
6 1 5- 2-
4 3 0

• Augment with 0 n1
• Vertices 2i, 2i-1 for each i

4
The Breakpoint Graph
11 2 1 9 10 3 4 8
7 6 5 0
6 1 5- 2-
4 3 0

• Augment with 0 n1
• Vertices 2i, 2i-1 for each i
• Blue edges between adjacent vertices

5
The Breakpoint Graph
11 2 1 9 10 3 4 8
7 6 5 0
6 1 5- 2-
4 3 0

• Augment with 0 n1
• Vertices 2i, 2i-1 for each i
• Blue edges between adjacent vertices
• Red edges between consecutive labels 2i,2i1

6
Sort a given breakpoint graph
11 2 1 9 10 3 4 8
7 6 5 0
into n1 trivial cycles
11 10 9 8 7 6 5 4
3 2 1 0
7
Sort a given breakpoint graph
11 2 1 9 10 3 4 8
7 6 5 0
into n1 trivial cycles
11 10 9 8 7 6 5 4
3 2 1 0
Conclusion We want to increase number of cycles
8
DefA reversal acts on two blue edges
11 2 1 9 10 3 4 8
7 6 5 0
cutting them and re-connecting them
11 2 1 9 10 3 4 7
8 6 5 0
9
A reversal can either
11 2 1 9 10 3 4 8
7 6 5 0
Act on two cycles, joining them (bad!!)
11 2 1 9 10 3 4 7
8 6 5 0
10
A reversal can either
11 2 1 9 10 3 4 8
7 6 5 0
Act on one cycle, changing it (profitless)
11 2 1 5 6 7 8
4 3 10 9 0
11
A reversal can either
11 2 1 9 10 3 4 8
7 6 5 0
Act on one cycle, splitting it (good move)
11 10 9 1 2 3 4 8
7 6 5 0
12
Basic Theorem
(Bafna, Pevzner 93)
Where dreversals needed (reversal
distance), and ccycles. Proof Every reversal
changes c by at most 1.
13
Basic Theorem
(Bafna, Pevzner 93)
Where dreversals needed (reversal
distance), and ccycles. Proof Every reversal
changes c by at most 1. Alternative formulation
where bbreakpoints, and c ignores short cycles
14
Oriented Edges
Red edges can be
Oriented
Right-to-Right
Left-to-Left
Unoriented
Left-to-Right
Right-to-Left
15
Oriented Edges
Red edges can be
Oriented
Right-to-Right
Left-to-Left
Unoriented
Left-to-Right
Right-to-Left
DefThis reversal acts on the red edge
16
Oriented Edges
Red edges can be
Oriented
Right-to-Right
Left-to-Left
Unoriented
Left-to-Right
Right-to-Left
DefThis reversal acts on the red edge
Thm A reversal acting on a red edge is good
the edge is oriented
17
Overlapping Edges
Def Two red edges are said to be overlapping if
they span intersecting intervals which do not
contain one another.
18
Overlapping Edges
Def Two red edges are said to be overlapping if
they span intersecting intervals which do not
contain one another The lines intersect
19
Overlapping Edges
Def Two red edges are said to be overlapping if
they span intersecting intervals which do not
contain one another The lines intersect
Thm A reversal acting on a red edge flips the
orientation of all edges overlapping it,
leaving other orientations unchanged
20
Overlapping Edges
Def Two red edges are said to be overlapping if
they span intersecting intervals which do not
contain one another The lines intersect
Thm if e,f,g overlap each other, then after
applying a reversal that acts on e, f and g do
not overlap
21
Overlap Graph
Nodes correspond to red edges. Two nodes are
connected by an arc if they overlap
22
Overlap Graph
Nodes correspond to red edges. Two nodes are
connected by an arc if they overlap
DefUnoriented connected components in the
overlap graph - all nodes correspond to oriented
edges.
23
Overlap Graph
Nodes correspond to red edges. Two nodes are
connected by an arc if they overlap

• DefUnoriented connected components in the
  overlap graph - all nodes correspond to oriented
  edges.
• Cannot be solved in only good moves

24
Dealing with Unoriented Components

• A profitless move on an oriented edge, making its
  component to oriented

25
Dealing with Unoriented Components

• A profitless move on an oriented edge, making its
  component to oriented
• or
• A bad move (reversal) joining cycles from
  different unoriented components, thus merging
  them flipping the orientation of many components
  on the way

26
Merging Unoriented Components
27
Merging Unoriented Components
28
Merging Unoriented Components
29
Merging Unoriented Components
30
Hurdles

• DefHurdle - an unoriented connected component
  which is consecutive along the cycle

31
Hurdles

• DefHurdle - an unoriented connected component
  which is consecutive along the cycle
• Thm
  (Hannenhalli, Pevzner 95)
• Proof A hurdle is destroyed by a profitless
  move, or
• at most two are destroyed (merged) by a bad move.

32
Hurdles

• DefHurdle - an unoriented connected component
  which is consecutive along the cycle
• Thm
  (Hannenhalli, Pevzner 95)
• Proof A hurdle is destroyed by a profitless
  move, or
• at most two are destroyed (merged) by a bad move.
• Thm