373 research outputs found
Distances and automatic sequences in distinguished variants of Hanoi graphs
In this thesis three open problems concerning Hanoi-type graphs are addressed. I prove a
theorem to determine all shortest paths between two arbitrary vertices s and t in the General SierpiĆski graph S_p^n with base p â„ 3 and exponent n â„ 0 and find an algorithm based on this theorem which gives us the index of the potential auxiliary subgraph, the distance between s and t and the best first move(s). Using the isomorphism between S_3^n and the Hanoi graphs H_3^n, this algorithm also determines the shortest paths in H_3^n. The results are also used in order to simplify proofs of already known metric properties of S_p^n. Additionally, I compute the average number of input pairs (s_i, t_i) for i Ï”{1,...,n} to be read by the algorithm. The Theorem and the algorithm for S_p^n are modified for the SierpiĆski triangle graphs, which are deeply connected to the well-known SierpiĆski triangle and the SierpiĆski graphs, with the result that the shortest paths in the SierpiĆski triangle graphs can be determined for the first time.
The Hanoi graphs H_3^n are then considered as directed graphs by differentiating the directions of the disc moves between the pegs of the corresponding Tower of Hanoi. For the problem to transfer a tower from one peg to another peg there are five different solvable variants. Here, the variants TH(C_3^+) and TH(K_3^-) are discussed concerning the infinite sequences of moves which arise from the solutions as n tends to infinity. The Allouche-Sapir Conjecture says that these sequences are not d-automatic for any d. I prove this for the TH(C_3^+) sequence with the aid of the frequency of a letter and its rationality in automatic sequences. For the TH(K_3^-) sequence I employ Cobhamâs Theorem about multiplicative independence, automatic sequences and ultimate periodicity. I show that this sequence is the image, under a 1-uniform morphism, of an iterative fixed point of a primitive prolongable endomorphism.
F. Durandâs methodá” is then used for the decision about the question whether the sequence is ultimately periodic. The method of I. V. Mitrofanová”, which works with subword schemata,is applied to the problem as well. Using the theory of recognisable sets, a sufficient condition for deciding the question about the automaticity of the TH(K_3^-) sequence is deduced.
Finally, a yet not studied distance problem on the so-called Star Tower of Hanoi, which is based on the star graph S t(4), is considered. Assuming that the Frame-Stewart type strategy is optimal, a recurrence for the length of the resulting paths is deduced and solved up to n = 12.
á” F. Durand, HD0L Ï-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012
á” I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 2015Ich beweise ein Theorem zur Bestimmung aller kĂŒrzesten Wege zwischen zwei beliebigen
Ecken s und t in den allgemeinen SierpiĆski-Graphen S_p^n mit Basis p â„ 3 und Exponent
n â„ 0 und erstelle auf diesem Theorem beruhend einen Algorithmus, der den Index des allfĂ€lligen Hilfsuntergraphen, den Abstand zwischen s und t und einen besten ersten Schritt liefert. Unter Verwendung des Isomorphismus zwischen S_3^n und den Hanoi-Graphen H_3^n bestimmt dieser Algorithmus auch die kĂŒrzesten Wege in H_3^n. Die Ergebnisse werden benutzt, um Beweise bereits bekannter metrischer Eigenschaften der S_p^n zu vereinfachen. ZusĂ€tzlich berechne ich die durchschnittlich benötigte Anzahl von Eingabepaaren (s_i, t_i) fĂŒr i Ï”{1,...,n} in den Algorithmus. Das Theorem und der Algorithmus fĂŒr S_p^n werden fĂŒr die Klasse der SierpiĆski-Dreiecksgraphen, welche in direktem Zusammenhang mit dem berĂŒhmten SierpiĆski-Dreieck und den SierpiĆski-Graphen stehen, modifiziert, sodass erstmals auch die kĂŒrzesten Wege in diesen Graphen bestimmt werden können.
Die Hanoi-Graphen H_3^n werden dann als gerichtete Graphen betrachtet, indem man die Richtungen der Bewegungen zwischen den StĂ€ben des entsprechenden Turms von Hanoi differenziert. FĂŒr das Problem des Versetzens eines Turms von einem Stab auf einen anderen gibt es fĂŒnf verschiedene lösbare Varianten. Die Varianten TH(C_3^+) und TH(K_3^-) werden bezĂŒglich der unendlichen Folgen von Bewegungen betrachtet, die sich durch die Lösung fĂŒr n gegen Unendlich strebend ergeben. Die Allouche-Sapir-Vermutung besagt, dass fĂŒr kein d diese Folgen d-automatisch erzeugt sind. Ich beweise dies fĂŒr die TH(C_3^+) Folge mit Hilfe der Theorie ĂŒber die HĂ€ufigkeit eines Buchstabens und deren RationalitĂ€t in automatisch erzeugten Folgen. FĂŒr die TH(K_3^-) Folge wird Cobhams Theorem ĂŒber multiplikative UnabhĂ€ngigkeit, automatisch erzeugte Folgen und ultimative PeriodizitĂ€t verwendet. Ich zeige, dass diese Folge das Bild, unter einem 1-uniformen Morphismus, eines iterativen Fixpunktes eines primitiven verlĂ€ngerbaren Endomorphismus ist. Die Methode von F. Durandá” wird dann fĂŒr die Entscheidung ĂŒber die Frage, ob die Folge ultimativ periodisch ist, verwendet. Ebenso wird die Methode von I. V. Mitrofanová”, welche mit Teilwortschemata arbeitet, auf das Problem angewandt. Unter Verwendung der Theorie ĂŒber erkennbare Mengen wird eine hinreichende Bedingung fĂŒr die Frage der AutomatizitĂ€t der TH(K_3^-) Folge hergeleitet.
Zuletzt wird ein bislang nicht untersuchtes Abstandsproblem im sogenannten Stern-Turm-von-
Hanoi betrachtet, welcher auf dem Stern-Graphen St(4) beruht. Unter der Annahme, dass die Frame-Stewart-Strategie optimal sei, wird eine Rekursionsvorschrift fĂŒr die LĂ€nge der so gewonnenen Wege entwickelt und bis n = 12 gelöst.
á” F. Durand, HD0L Ï-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012
á” I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 201
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Towers of Hanoi and generalized Towers of Hanoi
We investigate the time and space used by algorithms which solve the Towers of Hanoi problem
Additive Pattern Database Heuristics
We explore a method for computing admissible heuristic evaluation functions
for search problems. It utilizes pattern databases, which are precomputed
tables of the exact cost of solving various subproblems of an existing problem.
Unlike standard pattern database heuristics, however, we partition our problems
into disjoint subproblems, so that the costs of solving the different
subproblems can be added together without overestimating the cost of solving
the original problem. Previously, we showed how to statically partition the
sliding-tile puzzles into disjoint groups of tiles to compute an admissible
heuristic, using the same partition for each state and problem instance. Here
we extend the method and show that it applies to other domains as well. We also
present another method for additive heuristics which we call dynamically
partitioned pattern databases. Here we partition the problem into disjoint
subproblems for each state of the search dynamically. We discuss the pros and
cons of each of these methods and apply both methods to three different problem
domains: the sliding-tile puzzles, the 4-peg Towers of Hanoi problem, and
finding an optimal vertex cover of a graph. We find that in some problem
domains, static partitioning is most effective, while in others dynamic
partitioning is a better choice. In each of these problem domains, either
statically partitioned or dynamically partitioned pattern database heuristics
are the best known heuristics for the problem
Front-to-End Bidirectional Heuristic Search with Near-Optimal Node Expansions
It is well-known that any admissible unidirectional heuristic search
algorithm must expand all states whose -value is smaller than the optimal
solution cost when using a consistent heuristic. Such states are called "surely
expanded" (s.e.). A recent study characterized s.e. pairs of states for
bidirectional search with consistent heuristics: if a pair of states is s.e.
then at least one of the two states must be expanded. This paper derives a
lower bound, VC, on the minimum number of expansions required to cover all s.e.
pairs, and present a new admissible front-to-end bidirectional heuristic search
algorithm, Near-Optimal Bidirectional Search (NBS), that is guaranteed to do no
more than 2VC expansions. We further prove that no admissible front-to-end
algorithm has a worst case better than 2VC. Experimental results show that NBS
competes with or outperforms existing bidirectional search algorithms, and
often outperforms A* as well.Comment: Accepted to IJCAI 2017. Camera ready version with new timing result
Algorithmic Puzzles: History, Taxonomies, and Applications in Human Problem Solving
The paper concerns an important but underappreciated genre of algorithmic puzzles, explaining what these puzzles are, reviewing milestones in their long history, and giving two different ways to classify them. Also covered are major applications of algorithmic puzzles in cognitive science research, with an emphasis on insight problem solving, and the advantages of algorithmic puzzles over some other classes of problems used in insight research. The author proposes adding algorithmic puzzles as a separate category of insight problems, suggests 12 specific puzzles that could be useful for research in insight problem solving, and outlines several experiments dealing with other cognitive aspects of solving algorithmic puzzles
Distances and automatic sequences in distinguished variants of Hanoi graphs
In this thesis three open problems concerning Hanoi-type graphs are addressed. I prove a
theorem to determine all shortest paths between two arbitrary vertices s and t in the General SierpiĆski graph S_p^n with base p â„ 3 and exponent n â„ 0 and find an algorithm based on this theorem which gives us the index of the potential auxiliary subgraph, the distance between s and t and the best first move(s). Using the isomorphism between S_3^n and the Hanoi graphs H_3^n, this algorithm also determines the shortest paths in H_3^n. The results are also used in order to simplify proofs of already known metric properties of S_p^n. Additionally, I compute the average number of input pairs (s_i, t_i) for i Ï”{1,...,n} to be read by the algorithm. The Theorem and the algorithm for S_p^n are modified for the SierpiĆski triangle graphs, which are deeply connected to the well-known SierpiĆski triangle and the SierpiĆski graphs, with the result that the shortest paths in the SierpiĆski triangle graphs can be determined for the first time.
The Hanoi graphs H_3^n are then considered as directed graphs by differentiating the directions of the disc moves between the pegs of the corresponding Tower of Hanoi. For the problem to transfer a tower from one peg to another peg there are five different solvable variants. Here, the variants TH(C_3^+) and TH(K_3^-) are discussed concerning the infinite sequences of moves which arise from the solutions as n tends to infinity. The Allouche-Sapir Conjecture says that these sequences are not d-automatic for any d. I prove this for the TH(C_3^+) sequence with the aid of the frequency of a letter and its rationality in automatic sequences. For the TH(K_3^-) sequence I employ Cobhamâs Theorem about multiplicative independence, automatic sequences and ultimate periodicity. I show that this sequence is the image, under a 1-uniform morphism, of an iterative fixed point of a primitive prolongable endomorphism.
F. Durandâs methodá” is then used for the decision about the question whether the sequence is ultimately periodic. The method of I. V. Mitrofanová”, which works with subword schemata,is applied to the problem as well. Using the theory of recognisable sets, a sufficient condition for deciding the question about the automaticity of the TH(K_3^-) sequence is deduced.
Finally, a yet not studied distance problem on the so-called Star Tower of Hanoi, which is based on the star graph S t(4), is considered. Assuming that the Frame-Stewart type strategy is optimal, a recurrence for the length of the resulting paths is deduced and solved up to n = 12.
á” F. Durand, HD0L Ï-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012
á” I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 2015Ich beweise ein Theorem zur Bestimmung aller kĂŒrzesten Wege zwischen zwei beliebigen
Ecken s und t in den allgemeinen SierpiĆski-Graphen S_p^n mit Basis p â„ 3 und Exponent
n â„ 0 und erstelle auf diesem Theorem beruhend einen Algorithmus, der den Index des allfĂ€lligen Hilfsuntergraphen, den Abstand zwischen s und t und einen besten ersten Schritt liefert. Unter Verwendung des Isomorphismus zwischen S_3^n und den Hanoi-Graphen H_3^n bestimmt dieser Algorithmus auch die kĂŒrzesten Wege in H_3^n. Die Ergebnisse werden benutzt, um Beweise bereits bekannter metrischer Eigenschaften der S_p^n zu vereinfachen. ZusĂ€tzlich berechne ich die durchschnittlich benötigte Anzahl von Eingabepaaren (s_i, t_i) fĂŒr i Ï”{1,...,n} in den Algorithmus. Das Theorem und der Algorithmus fĂŒr S_p^n werden fĂŒr die Klasse der SierpiĆski-Dreiecksgraphen, welche in direktem Zusammenhang mit dem berĂŒhmten SierpiĆski-Dreieck und den SierpiĆski-Graphen stehen, modifiziert, sodass erstmals auch die kĂŒrzesten Wege in diesen Graphen bestimmt werden können.
Die Hanoi-Graphen H_3^n werden dann als gerichtete Graphen betrachtet, indem man die Richtungen der Bewegungen zwischen den StĂ€ben des entsprechenden Turms von Hanoi differenziert. FĂŒr das Problem des Versetzens eines Turms von einem Stab auf einen anderen gibt es fĂŒnf verschiedene lösbare Varianten. Die Varianten TH(C_3^+) und TH(K_3^-) werden bezĂŒglich der unendlichen Folgen von Bewegungen betrachtet, die sich durch die Lösung fĂŒr n gegen Unendlich strebend ergeben. Die Allouche-Sapir-Vermutung besagt, dass fĂŒr kein d diese Folgen d-automatisch erzeugt sind. Ich beweise dies fĂŒr die TH(C_3^+) Folge mit Hilfe der Theorie ĂŒber die HĂ€ufigkeit eines Buchstabens und deren RationalitĂ€t in automatisch erzeugten Folgen. FĂŒr die TH(K_3^-) Folge wird Cobhams Theorem ĂŒber multiplikative UnabhĂ€ngigkeit, automatisch erzeugte Folgen und ultimative PeriodizitĂ€t verwendet. Ich zeige, dass diese Folge das Bild, unter einem 1-uniformen Morphismus, eines iterativen Fixpunktes eines primitiven verlĂ€ngerbaren Endomorphismus ist. Die Methode von F. Durandá” wird dann fĂŒr die Entscheidung ĂŒber die Frage, ob die Folge ultimativ periodisch ist, verwendet. Ebenso wird die Methode von I. V. Mitrofanová”, welche mit Teilwortschemata arbeitet, auf das Problem angewandt. Unter Verwendung der Theorie ĂŒber erkennbare Mengen wird eine hinreichende Bedingung fĂŒr die Frage der AutomatizitĂ€t der TH(K_3^-) Folge hergeleitet.
Zuletzt wird ein bislang nicht untersuchtes Abstandsproblem im sogenannten Stern-Turm-von-
Hanoi betrachtet, welcher auf dem Stern-Graphen St(4) beruht. Unter der Annahme, dass die Frame-Stewart-Strategie optimal sei, wird eine Rekursionsvorschrift fĂŒr die LĂ€nge der so gewonnenen Wege entwickelt und bis n = 12 gelöst.
á” F. Durand, HD0L Ï-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy). Journal of Uniform Distribution Theory, 7(1):199-215, 2012
á” I. V. Mitrofanov, Periodicity of Morphic Words, Journal of Mathematical Sciences, 206(6):679-687, 201
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