The paper deals with the shortest path-based in-trees on a grid graph. There a root is supposed to move among all vertices. As such a spanning mobility pattern, root trajectories based on a Hamilton path or cycle are discussed. Along such a trajectory, each vertex randomly selects the next hop on the shortest path to the root. Under those assumptions, this paper shows that a root trajectory termed an S-path provides the minimum expected symmetric difference. Numerical experiments show that another trajectory termed a Right-cycle also provides the minimum result.