The number of moves of the largest disc in shortest paths on Hanoi graphs

Aumann, Simon; Götz, Katharina; Hinz, Andreas; Petr, Ciril

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Title:	The number of moves of the largest disc in shortest paths on Hanoi graphs
Authors:	ID Aumann, Simon (Author) ID Götz, Katharina (Author) ID Hinz, Andreas (Author) ID Petr, Ciril (Author)
Files:	Electronic_Journal_of_Combinatorics_2014_Aumann_et_al._The_number_of_moves_of_the_largest_disc_in_shortest_paths_on_Hanoi_graphs.pdf (376,70 KB) MD5: 54B714EE86235B1DDA9E49C4E1B94267 PID: 20.500.12556/dkum/18893bd4-6b0b-478d-a2db-1bf89baee89d http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p38
Language:	English
Work type:	Scientific work
Typology:	1.01 - Original Scientific Article
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs $H_p^n$ in a more general setting. Here $p$ stands for the number of pegs and $n$ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length $p-1$ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all $n \geqslant p(p-2)$ , of optimal paths where $p-1$ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.
Keywords:	graph theory, Tower of Hanoi, Hanoi graphs, shortest paths, symmetries, breadth-first search
Publication status:	Published
Publication version:	Version of Record
Year of publishing:	2014
Number of pages:	str. 1-22
Numbering:	Letn. 21, št. 4
PID:	20.500.12556/DKUM-67353
ISSN:	1077-8926
UDC:	519.17
ISSN on article:	1077-8926
COBISS.SI-ID:	17173081
NUK URN:	URN:SI:UM:DK:CUEEYWTE
Publication date in DKUM:	14.08.2017
Views:	1714
Downloads:	248
Metadata:
Categories:	Misc.
:	AUMANN, Simon, GÖTZ, Katharina, HINZ, Andreas and PETR, Ciril, 2014, The number of moves of the largest disc in shortest paths on Hanoi graphs. The Electronic journal of combinatorics [online]. 2014. Vol. 21, no. 4, p. 1–22. [Accessed 15 March 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=67353 Copy citation

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Record is a part of a journal

Title:	The Electronic journal of combinatorics
Shortened title:	Electron. j. comb.
Publisher:	N.J. Calkin and H.S. Wilf
ISSN:	1077-8926
COBISS.SI-ID:	6973785

Secondary language

Language:	Slovenian
Title:	Število premikov največje ploščice na najkrajših poteh v hanojskih grafih
Abstract:	Kljub širšemu zanimanju za Frame-Stewartovo domnevo o optimalnem številu potez v klasičnem problemu hanojskega stolpa z več kot tremi položaji, je to prva študija o najkrajših poteh v hanojskih grafih $H_p^n$ , kjer $p$ predstavlja število položajev in $n$ število ploščic, če graf interpretiramo kot graf stanj hanojskega stolpa. Študija se še posebej loti analize premikov največje ploščice. Vzorec teh premikov je zakodiran kot binarni niz dolžine $p-1$ in prirejen vsakemu paru začetnega in končnega stanja posebej. K analizi problema se pristopa tako analitično kot tudi numerično. Glavni teoretični dosežek je obstoj optimalnih poti za vse $n \geqslant p(p-2)$ , na katerih so nujni $p-1$ premiki največje ploščice. Numerični rezultati so pridobljeni z modificiranim algoritmom, zasnovanim na algoritmu iskanja v širino. V namene optimizacije iskanja se uporabijo simetrije grafov. Numerična evidenca vodi k nekaj domnevam o (ne)obstoju, ki jih teoretični rezultati ne pokrivajo in mogoče nam pomaga razkriti tudi kakšno skrivnost še nerazrešene Frame-Stewartove domneve.
Keywords:	teorija grafov, hanojski stolp, hanojski graf, najkrajša pot, simetričnosti, iskanje v širino

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