Exponential vs. Subexponential Tower of Hanoi Variants

We deal here with Tower of Hanoi variants played on digraphs. A major source for such variants is achieved by adding pegs and/or restricting direct moves between certain pairs of pegs. It is natural to represent a variant of this kind by a directed graph whose vertices are the pegs, and an arc from one vertex to another indicates that it is allowed to move a disk from the former peg to the latter, provided that the usual rules are not violated. We denote the number of pegs by h. For example, the variant with no restrictions on moves is represented by the Complete graph Kh; the variant in which the pegs constitute a cycle and moves are allowed only in one direction is represented by the uni-directional graph Cyclich . For all 3-peg variants, the number of moves grows exponentially fast with n. However, for h ≥ 4 pegs, this is not the case. For example, for Cyclich the number of moves is exponential for any h, while for a path on 4 vertices it is O( √ n3 √ ). This paper characterizes the graphs for which the transfer of a tower of size n of disks from a peg to another requires exponentially many moves as a function of n. To this end we introduce the notion of a shed, as a graph property. A vertex v in a strongly-connected directed graph G = (V, E) is a shed if the subgraph of G induced by V (G) − {v} contains a strongly connected subgraph on 3 or more vertices. Graphs with sheds will be shown to be much more efficient than those without sheds, for the particular domain of the Tower of Hanoi puzzle. Specifically, we show how, given a shed, we can indeed move a tower of disks from any peg to any other within O(λ α ) moves, where λ > 1 and α = 1 2 +o(1). For graphs without a shed, this is impossible. Submitted: March 2014 Reviewed: December 2015 Revised: July 2016 Accepted: August 2016 Final: October 2016 Published: October 2016 Article type: Regular paper Communicated by: M. Kaufmann Research supported in part by the Sapir Academic College, Israel E-mail addresses: [email protected] (Daniel Berend) [email protected] (Amir Sapir) 462 Berend and Sapir Exp. vs. Subexp. Tower of Hanoi Variants