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A bubble-sort graph is a variant of Cayley graphs and it is suitable as a topology for massively parallel systems because of its simple and regular structure. Therefore, we focus on nth bubble-sort graphs and propose an algorithm to obtain n-1 disjoint paths between arbitrary two nodes in the complexity of polynomial order of n which is the degree of the graph plus one. We estimate the time complexity of the algorithm and the sum of path lengths after proving the correctness of the algorithm. Moreover, we report the results of computer experiment to evaluate average performance of the algorithm.