The weighted Tower of Hanoi

TL;DR

This paper introduces the weighted Tower of Hanoi, a generalization that minimizes total move cost using an optimal dynamic algorithm, expanding understanding of Tower of Hanoi variants.

Contribution

It presents an optimal dynamic programming algorithm for the weighted Tower of Hanoi and explores its properties and relations to other variants.

Findings

Developed an optimal dynamic algorithm for weighted Tower of Hanoi.

Established properties and relations of the weighted problem with other variants.

Abstract

The weighted Tower of Hanoi is a new generalization of the classical Tower of Hanoi problem, where a move of a disc between two pegs $i$ and $j$ is weighted by a positive real $w_{ij}\geq 0$. This new problem generalizes the concept of finding the minimum number of moves to solve the Tower of Hanoi, to find a sequence of moves with the minimum total cost. We present an optimal dynamic algorithm to solve the weighted Tower of Hanoi problem, we also establish some properties of this problem, as well as its relation with the Tower of Hanoi variants that are based on move restriction.