Figure 2

Building a 8-trace with 4-traces. This schema shows a sequence of reversals that optimally sorts π⁰ = (−7, + 8,−3, + 2, + 6,−5,−1, + 4) into π⁸ = ( + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8). This sequence of reversals is represented by the 8-trace {1,−,6,8}{1,−,3,5,6}{2,5,6}{6}{8}|{1,−,7}{3,4}|{3,−,6}. This 8-trace can be obtained by the application of the reversals of the 4-trace C = {2,5,6}{3,4}{6}|{3,−,6} (it sorts π⁴into π⁸) to the sequence of reversals of the 4-trace A = {1,−,6,8}{1,−,3,5,6}{8}|{1,−,7} (it sorts π⁰into π⁴). The algorithm DFA can be used to obtain the two 4-traces which sort π⁰into π⁴and the two 4-traces which sort π⁴into π⁸. Combining these 4-traces, we obtain all 8-traces of solutions which sort π⁰into π⁸and have π⁴as an intermediary permutation.