Toward a Dynamic Programming Solution for the 4-peg Tower of Hanoi Problem with Configurations

TL;DR

This paper introduces a dynamic programming method using tabling in B-Prolog to analyze the 4-peg Tower of Hanoi problem with configurations, highlighting the need for adaptable algorithms for various disk arrangements.

Contribution

It presents a novel dynamic programming approach for the 4-peg Tower of Hanoi with configurations, contrasting it with existing algorithms and exploring their limitations.

Findings

Partitioning strategies vary with configurations.

Optimal partitions for classic problems may not suit configurations.

Some configurations may require new algorithms.

Abstract

The Frame-Stewart algorithm for the 4-peg variant of the Tower of Hanoi, introduced in 1941, partitions disks into intermediate towers before moving the remaining disks to their destination. Algorithms that partition the disks have not been proven to be optimal, although they have been verified for up to 30 disks. This paper presents a dynamic programming approach to this algorithm, using tabling in B-Prolog. This study uses a variation of the problem, involving configurations of disks, in order to contrast the tabling approach with the approaches utilized by other solvers. A comparison of different partitioning locations for the Frame-Stewart algorithm indicates that, although certain partitions are optimal for the classic problem, they need to be modified for certain configurations, and that random configurations might require an entirely new algorithm.