The Tower of Hanoi problem on Path_h graphs

TL;DR

This paper analyzes the Tower of Hanoi problem on linearly connected pegs, showing that even with fewer interconnections, the move count grows sub-exponentially with the number of disks.

Contribution

It provides a detailed analysis and tight upper bounds for the number of moves required in the Path_h variant of the Tower of Hanoi problem.

Findings

Number of moves grows sub-exponentially with disks

Identifies recursive task structures in Path_h graphs

Provides tight upper bounds depending on parameters

Abstract

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.