The Tower of Hanoi problem on Path_h graphs

The generalized Tower of Hanoi problem with h \ge 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path_h 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)/2 possible bi-directional interconnections among pegs, here there are only h-1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. 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, n and the source and destination pegs.

💡 Research Summary

The paper investigates a constrained variant of the classic Tower of Hanoi puzzle in which the pegs are arranged linearly (the Pathₕ graph) and a disk may be moved only to an adjacent peg. For h ≥ 4 pegs the unrestricted version (complete graph Kₕ) is known to require a sub‑exponential number of moves, roughly Θ(n^{c·n^{1/(h‑2)}}). The authors ask whether the drastic reduction of inter‑peg connections—from h(h‑1)/2 to h‑1—affects this growth rate.

The authors first formalize the problem. Pegs are numbered 1…h, and a configuration is a legal distribution of n distinct disks among the pegs. A “perfect configuration” R_{h,i,n} denotes all n disks stacked on peg i. The quantity |R_{h,i,n}→R_{h,j,n}| is the minimum number of moves needed to transfer the whole tower from peg i to peg j. The worst‑case distance over all source–destination pairs is defined as Path(h,n)=max_{i<j}|R_{h,i,n}→R_{h,j,n}|.

A key technical device is the notion of a block: a set of disks with consecutive sizes. Blocks are ordered by size, and a block A is “lighter” than block B if the largest disk of A is smaller than the smallest disk of B. This allows the authors to split a tower into three parts—small disks (B_s), large disks (B_l), and the single largest disk (B_max)—and treat each part recursively.

The paper’s main contributions are twofold. First, for the smallest non‑trivial case h = 4, the authors design an explicit algorithm called FourMove. FourMove proceeds in three stages:

1. Spread – recursively move the small block B_s from peg 1 to the farthest peg 4.
2. Circular shift – using only the middle three pegs (a Path₃ subgraph), rotate the large block B_l and the single largest disk so that their order is reversed. This is achieved by invoking a known optimal algorithm for Path₃, which needs 3^{m}−1 moves to transfer m disks between the two extreme pegs of a three‑peg line (or half that number for adjacent pegs).
3. Accumulate – reverse the process of Spread, gathering the blocks onto peg 4 in the correct order.

The algorithm chooses m = round(√(2n)) and defines B_s =