Finding genomic distance based on gene order is a classic problem in genome rearrangements. Efficient exact algorithms for genomic distances based on inversions and/or translocations have been found but are complicated by special cases, rare in simulations and empirical data. We seek a universal operation underlying a more inclusive set of evolutionary operations and yielding a tractable genomic distance with simple mathematical form. We study a universal double-cut-and-join operation that accounts for inversions, translocations, fissions and fusions, but also produces circular intermediates which can be reabsorbed. The genomic distance, computable in linear time, is given by the number of breakpoints minus the number of cycles (b-c) in the comparison graph of the two genomes; the number of hurdles does not enter into it. Without changing the formula, we can replace generation and re-absorption of a circular intermediate by a generalized transposition, equivalent to a block interchange, with weight two. Our simple algorithm converts one multi-linear chromosome genome to another in the minimum distance.
