Optimal algorithms for Tower of Hanoi problems with relaxed placement rules

Yefim Dinitz*, Shay Solomon

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations


We study generalizations of the Tower of Hanoi (ToH) puzzle with relaxed placement rules. In 1981, D. Wood suggested a variant, where a bigger disk may be placed higher than a smaller one if their size difference is less than k. In 1992, D. Poole suggested a natural disk-moving strategy, and computed the length of the shortest move sequence (algorithm) under its framework. However, other strategies were not considered, so the lower bound/optimality question remained open. In 1998, Beneditkis, Berend, and Safro were able to prove the optimality of Poole's algorithm for the first non-trivial case k=2 only. We prove it be optimal in the general case. Besides, we prove a tight bound for the diameter of the configuration graph of the problem suggested by Wood. Further, we consider a generalized setting, where the disk sizes should not form a continuous interval of integers. To this end, we describe a finite family of potentially optimal algorithms and prove that for any set of disk sizes, the best one among those algorithms is optimal. Finally, a setting with the ultimate relaxed placement rule (suggested by D. Berend) is defined. We show that it is not more general, by finding a reduction to the second setting.

Original languageEnglish
Title of host publicationAlgorithms and Computation - 17th International Symposium, ISAAC 2006, Proceedings
Number of pages12
StatePublished - 2006
Externally publishedYes
Event17th International Symposium on Algorithms and Computation, ISAAC 2006 - Kolkata, India
Duration: 18 Dec 200620 Dec 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4288 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference17th International Symposium on Algorithms and Computation, ISAAC 2006


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