  Genomic distance between two genomes, i.e., the smallest number of genome
rearrangements required to transform one genome into the other, is often used
as a measure of evolutionary closeness of the genomes in comparative genomics
studies. However, in models that include rearrangements of significantly
different "power" such as reversals (that are "weak" and most frequent
rearrangements) and transpositions (that are more "powerful" but rare), the
genomic distance typically corresponds to a transformation with a large
proportion of transpositions, which is not biologically adequate.
  Weighted genomic distance is a traditional approach to bounding the
proportion of transpositions by assigning them a relative weight {\alpha} > 1.
A number of previous studies addressed the problem of computing weighted
genomic distance with {\alpha} \leq 2.
  Employing the model of multi-break rearrangements on circular genomes, that
captures both reversals (modelled as 2-breaks) and transpositions (modelled as
3-breaks), we prove that for {\alpha} \in (1,2], a minimum-weight
transformation may entirely consist of transpositions, implying that the
corresponding weighted genomic distance does not actually achieve its purpose
of bounding the proportion of transpositions. We further prove that for
{\alpha} \in (1,2), the minimum-weight transformations do not depend on a
particular choice of {\alpha} from this interval. We give a complete
characterization of such transformations and show that they coincide with the
transformations that at the same time have the shortest length and make the
smallest number of breakages in the genomes.
  Our results also provide a theoretical foundation for the empirical
observation that for {\alpha} < 2, transpositions are favored over reversals in
the minimum-weight transformations.
