The breakpoint graph
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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

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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

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


Sort a given breakpoint graph

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


Sort a given breakpoint graph

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


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


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


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


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


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


Merging Unoriented Components


Merging Unoriented Components


Merging Unoriented Components


Merging Unoriented Components


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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