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2002, Expositiones Mathematicae
Hanoi graphs are the state graphs for Tower of Hanoi problems with three or more pegs. We prove hamiltonicity and present a complete analysis of planarity of these graphs.
ACM Transactions on Algorithms, 2008
The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed graph G with two distinguished vertices, S and D, and allowing moves only along arcs of this graph. An optimal solution for such a graph G is an algorithm that completes the task of moving a tower of any given number of disks from S to D in a minimal number of disk moves.
Discrete Applied Mathematics, 2012
The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.
2005
The Tower of Hanoi problem with h ≥ 4 pegs is long known to require a sub-exponential number of moves in order to transfer a pile of n disks from one peg to another. In this paper we discuss the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h − 1)/2 bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the task of moving n disks between any two pegs is still shown to grow sub-exponentially as a function of the number of disks.
Discrete Applied Mathematics, 2012
The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.
Lecture Notes in Computer Science, 2007
We study two aspects of a generalization of the Tower of Hanoi puzzle. In 1981, D. Wood suggested its 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 for this problem, but only in 2005, the authors proved it be optimal in the general case. We describe the family of all optimal solutions to this problem and present a closed formula for their number, as a function of the number of disks and k. Besides, we prove a tight bound for the diameter of the configuration graph of the problem suggested by Wood. Finally, we prove that the average length of shortest sequence of moves, over all pairs of initial and final configurations, is the same as the above diameter, up to a constant factor.
The Tower of Hanoi game is a classical puzzle in recreational mathematics, which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is 2n--1, to transfer a tower of n disks. But there are also other variations to the game, involving for example move edges weighted by real numbers. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three heaps.
International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2022
Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire has been considered on quite a few classes of graphs. Beeler and Gray introduced the natural idea of adding edges to make an unsolvable graph solvable. Recently, the graph invariant ms(G), which is the minimal number of additional edges needed to make G solvable, has been introduced and investigated on banana trees by the authors. In this article, we determine ms(G) for several families of unsolvable graphs. Furthermore, we provide some general results for this number of Hamiltonian graphs and graphs obtained via binary graph operations. I.
European Journal of Combinatorics, 2005
It is known that in the Tower of Hanoi graphs there are at most two different shortest paths between any fixed pair of vertices. A formula is given that counts, for a given vertex v, the number of vertices u such that there are two shortest u, v-paths. The formula is expressed in terms of Stern's diatomic sequence b(n) (n ≥ 0) and implies that only for vertices of degree two this number is zero. Plane embeddings of the Tower of Hanoi graphs are also presented that provide an explicit description of b(n) as the number of elements of the sets of vertices of the Tower of Hanoi graphs intersected by certain lines in the plane.
The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and the second author [7] established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all values of n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all values of n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and the second author in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all values of n and k. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift , and the pancake problem . ⋆
BIT, 1985
We present the iterative solutions of the Towers of Hanoi problems (standard, cyclic, and generalized) using the program transformation methodology of Burstall-Darlington. We derive algorithms with minimal time x space requirements. Their correctness proofs are trivial, as ~asual when applying the program transformation technique.
Introduction This lecture is about the well-known Tower of Hanoi problem. The problem is discussed in many mathematical texts, and is often used in computing science and articial intelligence as an illustration of ecursion" as a problem-solving strategy. The discussion of the problem in The Magical Maze" by Ian Stewart is similar (although less deep) than that given here, so if there is anything you don't understand you might try looking there. The goal of this lecture is to bridge some of what you have learnt in MC1, in particular the use of induction but also graphs, with what you have done in MC2. I will use the Tower of Hanoi problem to explain the dierence between the WHAT", the HOW" and the WHY" of algorithm development and show how proof by induction is used to relate the WHAT" to the HOW", thus providing the WHY" in the context of this particular problem. 2 Problem Specication | the WHAT" The Tower of Hanoi problem comes from ...
Journal of Graph Algorithms and Applications, 2016
We deal here with Tower of Hanoi variants played on digraphs. A major source for such variants is achieved by adding pegs and/or restricting direct moves between certain pairs of pegs. It is natural to represent a variant of this kind by a directed graph whose vertices are the pegs, and an arc from one vertex to another indicates that it is allowed to move a disk from the former peg to the latter, provided that the usual rules are not violated. We denote the number of pegs by h. For example, the variant with no restrictions on moves is represented by the Complete graph K h ; the variant in which the pegs constitute a cycle and moves are allowed only in one direction is represented by the uni-directional graph Cyclic h. For all 3-peg variants, the number of moves grows exponentially fast with n. However, for h ≥ 4 pegs, this is not the case. For example, for Cyclic h the number of moves is exponential for any h, while for a path on 4 vertices it is O(√ n3 √ 2n). This paper characterizes the graphs for which the transfer of a tower of size n of disks from a peg to another requires exponentially many moves as a function of n. To this end we introduce the notion of a shed, as a graph property. A vertex v in a strongly-connected directed graph G = (V, E) is a shed if the subgraph of G induced by V (G) − {v} contains a strongly connected subgraph on 3 or more vertices. Graphs with sheds will be shown to be much more efficient than those without sheds, for the particular domain of the Tower of Hanoi puzzle. Specifically, we show how, given a shed, we can indeed move a tower of disks from any peg to any other within O(λ n α) moves, where λ > 1 and α = 1 2 + o(1). For graphs without a shed, this is impossible.
Theoretical Computer Science, 2006
The Tower of Hanoi problem is generalized in such a way that the pegs are located at the vertices of a directed graph G, and moves of disks may be made only along edges of G. Leiss obtained a complete characterization of graphs in which arbitrarily many disks can be moved from the source vertex S to the destination vertex D. Here we consider graphs which do not satisfy this characterization; hence, there is a bound on the number of disks which can be handled. Denote by g n the maximal such number as G varies over all such graphs with n vertices and S, D vary over the vertices.
Discrete Applied Mathematics, 2002
It is proved that seven different approaches to the multi-peg Tower of Hanoi problem are all equivalent. Among them the classical approaches of Stewart and Frame from 1941 can be found.
Czechoslovak Mathematical Journal, 1997
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Theoretical Computer Science, 2013
The "Towers of Hanoi" is a problem that has been extensively studied and frequently generalized. We are interested in its generalization to arbitrary directed graphs and ask how many moves are required in a given graph to move n disks from the starting peg to the destination peg. Not all directed graphs allow solving this problem; we will call those graphs that do Hanoi graphs. We settle the question of what are the Hanoi graphs that require the largest number of moves.
Lecture Notes in Computer Science, 2006
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 diskmoving 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.
We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.
Journal of Interdisciplinary Mathematics, 2006
Two major generalized methods for solving the multi-peg tower of Hanoi problem are considered. These are the dynamic approach of the multi-peg problem as noted in Majumdar [8], and generalized recursive optimal solution for the multi-peg tower of Hanoi by Ikpotokin et al. [3]. It is also shown that the DP approach will utilize more storage space, more number of arithmetic operations and off course more time compare to the second method for the same number of peg t and disk, n.
Expositiones Mathematicae, 2005
The Hanoi graphs H n p model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling numbers of the second kind and Stern's diatomic sequence appear naturally in the graphs H n p. In this note, second-order Eulerian numbers and Lah numbers are added to this list. Considering a variant of the p-pegs n-discs problem, Catalan numbers are also encountered.
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