Minimum Movement of a Robot for Sorting on a Cycle

In a graph   with  vertices, there is an unique box which is finally laid on each vertex. Thus each vertex and box is both numbered from 1 to  and the box  should be laid on the vertex  . But, the box  is initially located on the vertex  according to a permutation  . In each step, the robot can walk along an edge of  and can carry at most one box at a time. Also when arriving at a vertex, the robot can swap the box placed there with the box it is carrying. The problem is to minimize the total step so that every vertex has its own box, that is, the shuffled boxes are sorted. In this paper, we shall find an upper bound of the minimum number of steps and show that the movement of the robot is found in    time when  is a cycle.