Which Multi-peg Tower of Hanoi Problems Are Exponential?

Connectivity properties are very important characteristics of a graph. Whereas it is usually referred to as a measure of a graph’s vulnerability, a relatively new approach discusses a graph’s

average connectivity

as a measure for the graph’s performance in some areas, such as communication. This paper deals with Tower of Hanoi variants played on digraphs, and proves they can be grouped into two categories, based on a certain connectivity attribute to be defined in the sequel.

A major source for Tower of Hanoi 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

K

h

graph; the variant in which the pegs constitute a cycle and moves are allowed only in one direction — by the uni-directional graph

Cyclic

h

.

For all 3-peg variants, the number of moves grows exponentially fast with

n

. However, for

h

 ≥ 4 peg variants, this is not the case. Whereas for

Cyclic

h

the number of moves is exponential for any

h

, for most of the other graphs it is sub-exponential. For example, for a path on 4 vertices it is

$O(\sqrt{n}3^{\sqrt{2n}})$

, for

n

disks.

This paper presents a necessary and sufficient condition for a graph to be an H-subexp, i.e., a graph for which the transfer of

n

disks from a peg to another requires sub-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

 − {

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 graph with a shed, we can indeed move a tower of

n

disks from any peg to any other within

O

(2

εn

) moves, where

ε

 > 0 is arbitrarily small.