Kombinatorika posplošenih Hanojskih stolpov

doktorska disertacija

Ciril Petr (Author), Sandi Klavžar (Mentor), Uroš Milutinović (Co-mentor)

Abstract

Vpeljemo popoln opis stanja posplošenih Hanojskih stolpov in delni opis, s katerim opišemo le razmestitev vrhnjih ploščic. Definiramo preslikavo iz popolnega v delni opis, ugotavljamo njeno surjektivnost, injektivnost, preštejemo elemente v sliki te preslikave, to je vse različne delne opise, računamo moč praslik, navedemo pogoj, kdaj delnemu opisu ustreza enoličen popolni opis, in preštejemo vse take delne opise stanj. Definiramo graf stanj posplošenih Hanojskih stolpov. Ogledamo si nekatere inducirane podgrafe. Na dva načina preštejemo vse povezave v grafu, preštejemo tudi število prestavitev posamezne ploščice ter izračunamo minimalno, maksimalno in povprečno stopnjo grafa. Definiramo pet strategij reševanja problema posplošenih Hanojskih stolpov, med katerimi sta tudi domnevno optimalni Framova in Stewartova strategija. Dokažemo, da so enakovredne glede na število premikov ploščic. Dokažemo obstoj in opišemo vse 1-popolne kode v grafih Sierpińskega, ki predstavljajo grafe stanj posplošenih Hanojskih stolpov s spremenjenim pravilom prestavljanja ploščic. Ta rezultat je posplošitev znanih rezultatov o grafih Hanojskih stolpov s tremi položaji, pri katerih pa je pristop bistveno drugačen. Podamo tudi optimalen dekodirni algoritem, ki za dano 1-popolno kodo in točko grafa ugotovi, ali je kodna točka. Če ni, poišče njej najbližjo kodno točko.

Keywords

matematika;računalništvo;kombinatorika;Hanojski stolpi;algoritem;najkrajša pot;grafi Sierpińskega;1-popolna koda;

Data

Language:	Slovenian
Year of publishing:	2004
Typology:	2.08 - Doctoral Dissertation
Organization:	UM PEF - Faculty of Education
Publisher:	[C. Petr]
UDC:	519.1:004(043.3)
COBISS:	13020761
Views:	869
Downloads:	28
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary title:	Combinatorics of generalized Towers of Hanoi
Secondary abstract:	We introduce a complete description of the state of generalized Towers of Hanoi, and partial description in which only positions of the top-most discs are specified. We define a mapping from the complete to the incomplete description, analyze its surjectivity and injectivity, count the elements in the image of this map, i.e. all the different partial descriptions, compute the cardinality of the preimages, give the condition for a partial description to have the unique complete description, and count all such partial descriptions. We define a state graph of generalized Towers of Hanoi. We look at some of the induced subgraphs. We count the number of edges in the graph in two different ways. We also count the number of moves of a certain disc, and calculate the minimum, maximum and average degree of the graph. Wedefine five strategies for solving the generalized Towers of Hanoi problem, including the presumed optimal strategies of Frame and Stewart. We prove that they are equivalent with respect to the number of discs moves. We prove the existence and describe all 1-perfect codes in Sierpiński graphs, which represent the state graphs of the generalized Towers of Hanoi with modified rules for moving discs. This result is a generalization of previously known results about the graphs of Towers of Hanoi with three pegs, where the approach is intrinsically different. We also present the optimal decoding algorithm, which for a given 1-perfect code and a vertex of a graph decides whether it is a code vertex, and if not, find its nearest code vertex.
Secondary keywords:	mathematics;computer science;combinatorics;Towers of Hanoi;algorithm;shortest path;Sierpiński graphs;1-perfect code;
URN:	http://www.dlib.si/?urn=URN:NBN:SI:doc-FCUAES0E
Type (COBISS):	Dissertation
Thesis comment:	Univerza v Mariboru, Pedagoška fakulteta, Oddelek za matematiko in računalništvo
Pages:	102 str.
ID:	1725143