The Breakpoint Graph

The Breakpoint Graph Augment with 0 = n

The Breakpoint Graph Augment with 0 = n+1 Vertices 2i, 2i-1 for each i

The Breakpoint Graph Augment with 0 = n+1 Vertices 2i, 2i-1 for each i Blue edges between adjacent vertices

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

into n+1 trivial cycles Sort a given breakpoint graph

Sort a given breakpoint graph Conclusion: We want to increase number of cycles into n+1 trivial cycles

Def:A reversal acts on two blue edges cutting them and re-connecting them

A reversal can either Act on two cycles, joining them (bad!!)

A reversal can either Act on one cycle, changing it (profitless)

A reversal can either Act on one cycle, splitting it (good move)

Basic Theorem Where d=#reversals needed (reversal distance), and c=#cycles. Proof: Every reversal changes c by at most 1. (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 Basic Theorem (Bafna, Pevzner 93)

Right-to-Right Left-to-Left Left-to-Right Right-to-Left Red edges can be : Oriented { Unoriented { Oriented Edges

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

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

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

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 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

Overlap Graph Nodes correspond to red edges. Two nodes are connected by an arc if they overlap

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

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

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

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: