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3. Perform a DCJ operation (this can be done by splitting, merging trees and lazy reversals as described). The Problem of Chromosome Reincorporation in DCJ Sorting and Halving 21 Perfect DCJ scenarios. B´erard et al. [15] studied the problem of finding a scenario transforming genome Π into Γ that does not break a given set of common intervals. An interval in genome Π is a set of markers such that the subgraph of GΠ induced by their extremities is connected. Intervals of Π have zero or two borders – adjacencies such that one extremity is inside and one outside.

For example, the permutation π = (2, 4, 1, -3) has two normal forms representing sorting sequences: {1,3,4}{4} {1,2,3}{2,3} and {1,4}{2} {1,2}{3,4}. The two normal forms of the traces describes the entire set of 16 sequences in a compact way. Braga et al. [4,6] developed an algorithm (BR) to generate normal forms of traces. BR is also basically the ASSR-FW as described in Section 3, where solutions are represented as normal forms of traces, NFtraces. Their idea was based on the fact that each (i − 1)-NFtrace is a prefix for an i-NFtrace.

Any two vertices of V that are adjacent in some chromosome in G, other than xt and xh deriving from the same x, are connected by a black edge (thick lines in Figure 2(b)), and any two adjacent Advances on Genome Duplication Distances 29 vertices in H are connected by a gray edge (thin lines in Figure 2(b)). Notice that adjacencies to O are not represented. In the case of circular chromosomes, each vertex in V is incident to exactly one black and one gray edge, and thus the graph uniquely decomposes into c(G, H) disjoint cycles of alternating edge colors.

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Audio Watermarking Techniques by Kim

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