Fig. 2

In a, an example input collection \(\mathcal {S}\) of \(m=|\mathcal {S}|=4\) weighted strings (for \(k=3\)), where the end-point weights are highlighted in bold font. In b, the order of the strings is changed according to the permutation \(\pi =[1,4,2,3]\) and, as a result, the number of runs is reduced by 1 (the last run in string 4 is glued with the first run of string 2). Lastly, in c, it is shown that changing the orientation of string 3 (taking the reverse complement of the string and reversing the order of the \(k\)-mer weights) makes it possible to glue other two runs. Given that reducing the number of runs by \(m-1\) is the best achievable reduction, the number of runs in c is therefore the minimum for the original collection in a