5/4/2023 0 Comments Sequential search algorithm![]() Therefore, the cycle ratio À rc ð C Þ of any alternate cycle C in G can take on at most w max l max j E j 2 different values. all Similarly, edge P e 2 C w the weights ð e Þ denominator,, can take and on lengths at P most e 2 C l are ð w e Þ max, can integral, j E j take different on the at most l max j E j different values. The cycle ratio of any alternate cycle C has the form When numerator, values. Let w max 1⁄4 max e 2 E w ð e Þ and l max 1⁄4 max e 2 E l ð e Þ. It can be shown that this search process terminates in a (pseudo)polynomial number of iterations if the edge weights and lengths are all integers. This implies that after termination b à 1⁄4 À rc ð C Þ and the current C is a desired MRA cycle. Furthermore, we know that, with the current edge-weight transformation w 0 ð e Þ 1⁄4 w ð e Þ À À rc ð C Þ Á l ð e Þ, there is no alternate cycle with negative total weight after termination. To avoid confusion, we specify that the cycle ratio À rc ð C Þ is always calculated using the original edge weights in G without applying the edge-weight transformation (5). The correctness of this algorithm comes from the fact that an alternate cycle C always has zero total weight when all the edge weights are transformed by w ð e Þ 1⁄4 w ð e Þ À À rc ð C Þ Á l ð e Þ, after Step 3. , where it is used for finding a minimum ratio cycle in a directed graph. This sequential-search algorithm is adapted from Ahuja et al. Then, we can use this NWA cycle- detection algorithm repeatedly to determine b à using the following sequential-search algorithm, as shown in Fig. ![]() Now, suppose that we have an NWA cycle-detection algorithm that can determine whether there is an NWA cycle in G and, if there is, can extract such a cycle, both in polynomial time. In this case, MRA cycles are the same as the cycles with zero total edge weight. Since edge lengths are nonnegative, there exists an optimal b 1⁄4 b à so that, after the above edge-weight transformation, the resulting MRA cycles have a cycle ratio of zero. À rc ðÁÞ denotes the cycle ratio after the edge-weight transformation.
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