Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions

Abstract

We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.

Multiplicatively Exact Algorithms for Transformation and Reconstruction of Directed Path-Cycle Graphs with Repeated Edges

“…For the obtained directed structures A + and B + with equal content, find their shortest (in the number of operations) transformation, and among those, choose the shortest transformation of undirected structures A into B. The first can be done, for example, by a very particular case of the algorithm from [2] or by the algorithm from [11]. From this, obtain a minimal reduction of A * B to a ch-final form, and then execute Stages 4-5 of the algorithm.…”

“…Remark 4. The condition of equal content includes the absence of paralogs, so a breakpoint graph a + b is defined in Section 6 and in Section 7, in the same way as in Section 1 (see also [2] or a simpler version in [11]), but not the same as Section 2 or Section 5.…”

“…One of the initial arrangements is described above. Choose the second using the following algorithm [2] for construction of the shortest transformation of structures. Moving from the leaves to the root, consider a current vertex v and its brother v'.…”

“…A structure is any directed graph in which components are paths and cycles, and edges are assigned with names (in this respect, the graph is weighted) (see figures and examples in [1,2]). An edge is always considered in a pair with its name (but not with a direction).…”

“…A final graph (or a graph of a final form) is a graph consisting of isolated vertices and of cycles of length 2 (one edge which is labeled with a, and the other with b). In [2], this definition was extended to the case of unequal content without paralogs, but here we do not need this. Even in the general case, the transformation problem is equivalent (through a linear algorithm) to the problem of minimum reduction of a + b to a final form.…”