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Fig. 5 | Algorithms for Molecular Biology

Fig. 5

From: Heuristic algorithms for best match graph editing

Fig. 5

A Example for a colored digraph \((\vec {G},\sigma )\) in which the “locally” optimal (first) split does not result in a global optimal BMG editing. The minimal UR-cost equals 3 and is attained only for the partition \({\mathcal {V}}=\{ \{a_1,a_2,a_3,b_1,b_2,b_3\}, \{a_4,a_5,a_6,b_4,b_5,b_6\}\}\), which was verified by full enumeration of all partitions and Lemma 13. For this partition, \(U(\vec {G},{\mathcal {V}})\) comprises the three purple arcs. B The two (isomorphic) induced subgraphs obtained by applying the locally optimal partition \({\mathcal {V}}\). Each of them has a (global) optimal BMG editing cost of 4. Therefore, the overall symmetric difference of an edited digraph (using the initial split \({\mathcal {V}}\) as specified) comprises at least \(c(\vec {G},{\mathcal {V}})+2\cdot 4=11\) arcs. C An optimal editing removes the 8 green arcs and results in a digraph that is explained by the tree in D. The optimality of this solution was verified using an implementation of the ILP formulation for BMG editing given in [8]

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