Santa Claus’ Towers of Hanoi

Chen, Xiaomin; Tian, Bin; Wang, Lei

doi:10.1007/s00373-007-0705-4

Cited by 9 publications

(15 citation statements)

References 6 publications

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“…Beneditkis, Berend, and Safro [2] proved Poole's algorithm to be optimal for the first non-trivial case k = 2 only. Optimality of Poole's algorithm in the general case was proved independently by Dinitz and Solomon [6,8] and by Chen et al [5]. It was proved in [7] that there is more than one shortest path in the BTH graph between any pair of perfect configurations, for all k ≥ 2; also, a complete characterization of the set of all such shortest paths was given therein, complemented with a closed formula, depending on n and k, for the cardinality of this set.…”

Section: The Bottleneck Th Problem and Configuration Graphmentioning

confidence: 89%

On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem

Solomon¹,

Solomon²

2013

2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood, 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; when k = 1 we arrive at the classic TH problem. 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 established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all 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 Solomon from SOFSEM'07 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 n and k. To this end we provide a new proof that applies to all values of n > k. That is, our proof is not a patch over the previous proof of Dinitz and Solomon that is tailored only for the regime k < n ≤ 3k, but is rather a stronger proof that is based on different principles and deeper ideas.We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift [Albert and Atkinson 2002, König andLübbecke 2008] and the well-studied pancake problem [Gates and Papadimitriou 1979].

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Section: The Bottleneck Th Problem and Configuration Graphmentioning

confidence: 89%

On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem

Solomon¹,

Solomon²

2013

2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…Of particular interest is E(n, 1). Chen et al (2007), based on computer search, report that, E(n, 1) = S(n, 1) for 1  n  7, but E(8, 1) = 57 when the disc D6 is chosen as the evildoer. In a recent paper, Majumdar et al (2019) have given explicitly the scheme for E(8,1) when the disc D6 is an evildoer.…”

Section: The Problem and The Solutionmentioning

confidence: 99%

“…Over the past decades, the Tower of Hanoi problem has seen many generalizations, some of which have been reviewed by Majumdar (2012Majumdar ( , 2013 and Hinz et al (2013). Chen et al (2007) have introduced a new variant of the Tower of Hanoi problem with n  1 discs, which allows r ( 1) violations of the "divine rule". In the new variant, the problem is to shift the tower of n discs from the peg S to the peg D in minimum number of moves, where for (at most) r moves, some disc may be placed directly on top of a smaller one.…”

Section: Introductionmentioning

confidence: 99%

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The Tower of Hanoi Problem with Evildoer Discs

Majumdar¹

2020

J Bangladesh Acad Sci

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This paper deals with a variant of the classical Tower of Hanoi problem with n (  1) discs, of which r discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Denoting by E(n, r) the minimum number of moves required to solve the new variant, is given a scheme find the optimality equation satisfied by E(n, r). An explicit form of E(n, r) is then obtained.

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“…We also 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 for all values of n ≤ k and n > 3k, up to a constant factor. X. Chen et al [2] considered independently a few ToH problems, including the bottleneck ToH problem. They also suggested a proof of optimality of Poole's algorithm; it is based on another technical approach, and is not less difficult than our proof.…”

Section: Introductionmentioning

confidence: 99%

Optimal Algorithms for Tower of Hanoi Problems with Relaxed Placement Rules

Dinitz

Solomon

2006

Algorithms and Computation

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Abstract. 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.

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Santa Claus’ Towers of Hanoi

Cited by 9 publications

References 6 publications

On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem

On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem

The Tower of Hanoi Problem with Evildoer Discs

Optimal Algorithms for Tower of Hanoi Problems with Relaxed Placement Rules

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