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AuthorAumann, Simon
Götz, Katharina A. M.
Hinz, Andreas M., 1954-
Petr, Ciril
Title The number of moves of the largest disc in shortest paths on Hanoi graphs [Elektronski vir] / Simon Aumann ... [et al.]
Other titlesŠtevilo premikov največje ploščice na najkrajših poteh v hanojskih grafih
Type/contenttype of material article - component part
LanguageEnglish
Publication date2014
Physical descriptionP4.38 (22 str.)
Electronic resource characteristicsEl. članek
NotesOpis vira z dne 20. 11. 2014
Soavtorji: Katharina A. M. Götz, Andreas M. Hinz, Ciril Petr
Bibliografija: str. 21-22 (24 enot)
Abstract
Uncontrolled subject headingsteorija grafov / hanojski stolp / hanojski graf / najkrajša pot / simetričnosti / iskanje v širino / graph theory / Tower of Hanoi / Hanoi graphs / shortest paths / symmetries / breadth-first search
UDC519.17
Other class numbers
05C12
68W05
05-04 (MSC 2010)
URLhttp://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i4p38/pdf
SummaryKljub 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.
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.
COBISS.SI-ID17173081
See publication: TI=The Electronic journal of combinatorics [Elektronski vir] ISSN: 1077-8926.- Vol. 21, iss. 4 (2014), P4.38 (22 str.)

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