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The Breakpoint Graph

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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  1. The Breakpoint Graph 1 5- 2- 4 3

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

  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

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

  6. 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

  7. 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

  8. 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

  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 d=#reversals needed (reversal distance), and c=#cycles. Proof: Every reversal changes c by at most 1.

  13. 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

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

  31. 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.

  32. 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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