An Almost Exact Linear Complexity Algorithm of the Shortest Transformation of Chain-Cycle Graphs

A "genome structure" is a labeled directed graph with vertices of degree 1 or 2. A set of operations over such graphs is fixed, and each of the operations has a certain cost, a strictly positive number. The transformation problem consists in the following: for given structures a and b and given costs, find a minimum total cost sequence of operations transforming a into b ("the shortest transformation of a into b"). Each operation corresponds to an "event", the latter being a change in the graph caused by executing one of the operations over it. The possibility of assigning different costs is important in applications, since it allows to distinguish between frequent and rare events. Apparently, arbitrary costs make the problem NP-hard, which results in nontriviality of passing from one restriction on costs to another, if the problem is solved by a linear or at least polynomial algorithm (assuming that P is not equal NP). We propose a novel linear time and space algorithm which constructs a sequence of operations transforming a to b with total cost close or equal to the absolute minimum. Namely, if all the so-called DCJ operations have the same cost w and if deletions and insertions have costs either both larger or both less than w, then the algorithm outputs a transformation of a into b with the total cost differing from the absolute minimum by at most 2w (in the former case) or equal to it (in the latter case). In some cases, the algorithm outputs an exact solution, e.g., in the case of circular genome structures. The condition on the costs of deletions and insertions can be omitted (although the proof described below does not include this general case).