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The Breakpoint Graph. 1 5- 2- 4 3 . The Breakpoint Graph. 6 1 5- 2- 4 3 0. Augment with 0 = n+1. The Breakpoint Graph.

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The Breakpoint Graph

1 5- 2- 4 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 = n+1
- Vertices 2i, 2i-1 for each i

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 = n+1
- Vertices 2i, 2i-1 for each i
- Blue edges between adjacent vertices

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 = n+1
- Vertices 2i, 2i-1 for each i
- Blue edges between adjacent vertices
- Red edges between consecutive labels 2i,2i+1

11 2 1 9 10 3 4 8 7 6 5 0

into n+1 trivial cycles

11 10 9 8 7 6 5 4 3 2 1 0

11 2 1 9 10 3 4 8 7 6 5 0

into n+1 trivial cycles

11 10 9 8 7 6 5 4 3 2 1 0

Conclusion:We want to increase number of cycles

Def:A 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

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

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 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

Basic Theorem

(Bafna, Pevzner 93)

Where d=#reversals needed (reversal distance),

and c=#cycles.

Proof: Every reversal changes c by at most 1.

Basic Theorem

(Bafna, Pevzner 93)

Where d=#reversals needed (reversal distance),

and c=#cycles.

Proof: Every reversal changes c by at most 1.

Alternative formulation:

where b=#breakpoints, and c ignores short cycles

Oriented Edges

Red edges can be :

Oriented{

Right-to-Right

Left-to-Left

Unoriented{

Left-to-Right

Right-to-Left

Oriented Edges

Red edges can be :

Oriented{

Right-to-Right

Left-to-Left

Unoriented{

Left-to-Right

Right-to-Left

Def:This reversal acts on the red edge

Oriented Edges

Red edges can be :

Oriented{

Right-to-Right

Left-to-Left

Unoriented{

Left-to-Right

Right-to-Left

Def:This reversal acts on the red edge

Thm: A reversal acting on a red edge is good

the edge is oriented

Overlapping Edges

Def: Two red edges are said to be overlapping if they span intersecting intervals which do not contain one another.

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

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

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

Overlap Graph

Nodes correspond to red edges.

Two nodes are connected by an arc if they overlap

Def:Unoriented connected components in the overlap graph - all nodes correspond to oriented edges.

Overlap Graph

Nodes correspond to red edges.

Two nodes are connected by an arc if they overlap

Def:Unoriented connected components in the overlap graph - all nodes correspond to oriented edges.

Cannot be solved in only good moves

Dealing with Unoriented Components

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

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

Hurdles

- Def:Hurdle - an unoriented connected component which is consecutive along the cycle

Hurdles

- Def:Hurdle - 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.

Hurdles

- Def:Hurdle - 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:

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