Again, say for the sake of contradiction that we have an optimal completion for which a component C′ of B contains two or more -paths.
Say that is an optimal completion for which an odd π - path P1 and an even π - path P2 are contained in different components of B, neither of which is an even 2-chain.
In our opinion, the incomplete data update approach in the nearest prototype strategy is a special case of that in the optimal completion strategy regardless of fuzziness for the membership of each vector.
□ If pΠ,γ is even, then the breakpoint graph of an optimal completion must contain a maximum set of even-length 2 - chains.
Thus, by Theorem 3, to identify the solution space of DCJ-indel sorting (an open problem), we simply need to enumerate the construction of indels of an optimal completion.
Every π - path (Γ - path) embedding into a 3-chain of an optimal completion must have the same parity.
The breakpoint graph of an optimal completion cannot have one 2-chain joining two odd π - paths and another 2-chain joining two even π-paths.
If is an optimal completion, then a component C∗ of B can only contain two or more - paths if C∗ is a 2 - bracelet.
Say for the sake of contradiction that we have an optimal completion such that B contains 3-chains [ P1 : P2 : P3] and [ P4 : P5 : P6], where P1 and P4 are π - paths of opposite parity.
If instead p Π, γ is even, then δ = 0, and the argument for constructing an optimal completion proceeds similarly, except that no - paths will remain after forming a maximal collection of 2-bracelets, eliminating the need for Step 4. □ Given genomes, the following algorithm constructs an optimal completion in O(N) time.
