the breakpoint graph 1 5- 2- 4 3. the breakpoint graph augment with 0 = n+1 6 1 5- 2- 4 3 0
TRANSCRIPT
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The Breakpoint Graph
1 5 -2 -4 3
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The Breakpoint Graph
• Augment with 0 = n+1
6 1 5 -2 -4 3 0
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The Breakpoint Graph
• Augment with 0 = n+1• Vertices 2i, 2i-1 for each i
6 1 5 -2 -4 3 0
11 2 1 9 10 3 4 8 7 6 5 0
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The Breakpoint Graph
• Augment with 0 = n+1• Vertices 2i, 2i-1 for each i • Blue edges between adjacent vertices
6 1 5 -2 -4 3 0
11 2 1 9 10 3 4 8 7 6 5 0
12,2 1 ii
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The Breakpoint Graph
• 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
6 1 5 -2 -4 3 0
11 2 1 9 10 3 4 8 7 6 5 0
12,2 1 ii
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11 2 1 9 10 3 4 8 7 6 5 0
11 10 9 8 7 6 5 4 3 2 1 0
into n+1 trivial cycles
Sort a given breakpoint graph
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11 2 1 9 10 3 4 8 7 6 5 0
Sort a given breakpoint graph
Conclusion: We want to increase number of cycles
11 10 9 8 7 6 5 4 3 2 1 0
into n+1 trivial cycles
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Def:A reversal acts on two blue edges
cutting them and re-connecting them
11 2 1 9 10 3 4 88 77 6 5 0
11 2 1 9 10 3 4 77 88 6 5 0
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A reversal can either
Act on two cycles, joining them (bad!!)
11 2 1 9 10 3 4 88 77 6 5 0
11 2 1 9 10 3 4 77 88 6 5 0
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A reversal can either
Act on one cycle, changing it (profitless)
11 2 1 9 10 3 4 88 77 6 5 0
11 2 1 5 66 7 8 4 3 10 9 0
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A reversal can either
Act on one cycle, splitting it (good move)
11 2 1 9 10 3 4 88 77 6 5 0
11 10 9 1 2 3 4 88 77 6 5 0
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Basic Theorem
)(1)( cnd
Where d=#reversals needed (reversal distance),
and c=#cycles.
Proof: Every reversal changes c by at most 1.
(Bafna, Pevzner 93)
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)(1)( cnd
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
)()()( cbd
Basic Theorem(Bafna, Pevzner 93)
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Right-to-Right
Left-to-Left
Left-to-Right
Right-to-Left
Red edges can be :
Oriented{
Unoriented{
Oriented Edges
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Right-to-Right
Left-to-Left
Left-to-Right
Right-to-Left
Red edges can be :
Oriented{
Unoriented{Def:This reversal acts on the red edge
Oriented Edges
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Right-to-Right
Left-to-Left
Left-to-Right
Right-to-Left
Red edges can be :
Oriented{
Unoriented{Def:This reversal acts on the red edge
Oriented Edges
Thm: A reversal acting on a red edge is good the edge is oriented
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Def: Two red edges are said to be overlapping if they span intersecting intervals which do not contain one another.
Overlapping Edges
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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
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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
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Thm: if e,f,g overlap each other, then after applying a reversal that acts on e, f and g do not overlap
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
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Overlap Graph
Nodes correspond to red edges.Two nodes are connected by an arc if they overlap
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Overlap Graph
Def:Unoriented connected components in the overlap graph - all nodes correspond to oriented edges.
Nodes correspond to red edges.Two nodes are connected by an arc if they overlap
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Overlap Graph
Def:Unoriented connected components in the overlap graph - all nodes correspond to oriented edges.
Cannot be solved in only good moves
Nodes correspond to red edges.Two nodes are connected by an arc if they overlap
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Dealing with Unoriented Components
• A profitless move on an oriented edge, making its component to oriented
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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
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Merging Unoriented Components
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Merging Unoriented Components
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Merging Unoriented Components
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Merging Unoriented Components
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Hurdles
• Def:Hurdle - an unoriented connected component which is consecutive along the cycle
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Hurdles
d b c h( ) ( ) ( ) ( )
• 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.
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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:
d b c h( ) ( ) ( ) ( )
d b c h f f( ) ( ) ( ) ( ) ( ), ( ) { , } 0 1