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.

通常通过假设所有倒置发生的可能性均等，来测量倒置过程中两个细菌基因组之间的距离。最近，引入了一种使用群论来计算反演距离的方法，该方法对于仅发生非常短的反演的模型是有效的。在本文中，我们展示了如何使用组理论框架为反转集中的任何加权建立最小距离，并概括了先前的方法。为此，我们使用组重写系统的理论，并利用Knuth-Bendix算法，这是该理论首次被引入基因组重排问题。该方法的中心思想是使用现有的群体理论方法在基因组空间中找到两个基因组之间的初始路径（例如，仅使用短反转），