Measuring the distance between two bacterial genomes under the inversion
process is usually done by assuming all inversions to occur with equal
probability. Recently, an approach to calculating inversion distance using
group theory was introduced, and is effective for the model in which only very
short inversions occur. In this paper, we show how to use the group-theoretic
framework to establish minimal distance for any weighting on the set of
inversions, generalizing previous approaches. To do this we use the theory of
rewriting systems for groups, and exploit the Knuth--Bendix algorithm, the
first time this theory has been introduced into genome rearrangement problems.
The central idea of the approach is to use existing group theoretic methods
to find an initial path between two genomes in genome space (for instance using
only short inversions), and then to deform this path to optimality using a
confluent system of rewriting rules generated by the Knuth--Bendix algorithm.