The Tower of Hanoi problem on Pathh graphs

The generalized Tower of Hanoi problem with h≥4h≥4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of nn disks from one peg to another. In this paper we study the Pathh variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only.Whereas in the simple variant there are h(h−1)/2h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h−1h−1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of nn disks between any two pegs also grows sub-exponentially as a function of nn. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h,nh,n and the source and destination pegs.