The Tower of Hanoi problem on Path_h graphs

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.

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.

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 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 . ⋆

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.

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.

International Journal of Computer Mathematics, 1995

We examine a variation of the famous Tower of Hanoi puzzle posed but not solved in a 1944 paper by Scorer et al. [5]. In this variation, disks of adjacent sizes can be exchanged, provided that they are at the top of their respective stacks. We present an algorithm for solving this variation, analyze its performance, and prove that it is optimal. Several exercises are listed at the end, ranging in difficulty from elementary to research level.

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.