Cited By
View all- Leiss E(2013)The worst Hanoi graphsTheoretical Computer Science10.1016/j.tcs.2013.05.020498(100-106)Online publication date: 1-Aug-2013
On an infinite family of solvable Hanoi graphs
Article No.: 13, Pages 1 - 22
Abstract
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.
In this article we present an algorithm which solves the problem for two infinite families of graphs, and prove its optimality. To the best of our knowledge, this is the first optimality proof for an infinite family of graphs.
Furthermore, we present a unified algorithm that solves the problem for a wider family of graphs and conjecture its optimality.
References
[1]
Allouche, J. P., Astoorian, D., Randall, J., and Shallit, J. O. 1994. Morphisms, squarefree strings, and the Tower of Hanoi puzzle. Amer. Math. Monthly 101, 651--658.
[2]
Azriel, D., and Berend, D. 2006. On a question of Leiss regarding the Hanoi Tower problem. Theor. Comput. Sci. 369, 1-3, 377--383.
[3]
Chen, X., and Shen, J. 2004. On the Frame-Stewart conjecture about the Towers Of Hanoi. Siam J. Comput. 33, 584--589.
[4]
Dinitz, Y., and Solomon, S. 2006. Optimal algorithms for Tower of Hanoi problems with relaxed placement rules. In Proceedings of the 17th International Symposium on Algorithms and Computation (ISAAC), T. Asano, Ed. Lecture Notes in Computer Science, vol. 4288, Springer, 36--47.
[5]
Dinitz, Y., and Solomon, S. 2008. Optimality of an algorithm solving the Bottleneck Tower of Hanoi problem. ACM Trans. Algor. 4, 3, Art. # 15.
[6]
Dudeney, H. E. 1907. The Canterbury Puzzles (and Other Curious Problems). Thomas Nelson and Sons, London.
[7]
Er, M. C. 1985. The complexity of the generalised cyclic Towers of Hanoi. J. Algor. 6, 351--358.
[8]
Frame, J. S. 1941. Solution to advanced problem 3918. Amer. Math. Monthly 48, 216--217.
[9]
Hinz, A. M., 1992. Pascal's triangle and the Tower of Hanoi. Amer. Math. Monthly 99, 538--544.
[10]
Korf, R. E., and Felner, A. 2007. Recent progress in heuristic search: A case study of the four-peg Towers of Hanoi problem. In Proceedings of the 20th International Joint Conferences on Artificial Intelligence (IJCAI), M. M. Veloso, Ed. International Joint Conferences on Artificial Intelligence, Menlo Park, CA, 2324--2329.
[11]
Leiss, E. L. 1983. Solving the ‘Towers of Hanoi’ on graphs. J. Combin. Inf. Syst. Sci. 8, 81--89.
[12]
Leiss, E. L. 1984. Finite Hanoi problems: How many discs can be handled? Congr. Numer. 44, 221--229.
[13]
Lucas, E. 1893. Récréations Mathématiques. Vol. III. Gauthier-Villars, Paris.
[14]
Lunnon, W. F. 1986. The Reve's puzzle. The Comput. J. 29, 478.
[15]
Minsker, S. 1989. The Towers of Hanoi rainbow problem: Coloring the rings. J. Algor. 10, 1--19.
[16]
Poole, D. 1992. The Bottleneck Towers of Hanoi problem. J. Recreational Math. 24, 3, 203--207.
[17]
Sapir, A. 2004. The Towers of Hanoi with forbidden moves. The Comput. J. 47, 20--24.
[18]
Scorer, R. S., Grundy, P. M., and Smith, C. A. B. 1944. Some binary games. The Math. Gazette 280, 96--103.
[19]
Stewart, B. M. 1939. Advanced problem 3918. Amer. Math. Monthly 46, 363.
[20]
Stewart, B. M. 1941. Solution to advanced problem 3918. Amer. Math. Monthly 48, 217--219.
[21]
Stockmeyer, P. K. 1994. Variations on the four-post Tower of Hanoi puzzle. Congr. Numer. 102, 3--12.
[22]
Szegedy, M. 1999. In how many steps the k peg version of the Towers of Hanoi game can be solved? In Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science (STACS), C. Meinel and S. Tison, Eds. Lecture Notes in Computer Science, vol. 1563, Springer, 356--361.
[23]
Wood, D. 1981. The Towers of Brahma and Hanoi revisited. J. Recreational Math. 14, 1, 17--24.
Index Terms
- On an infinite family of solvable Hanoi graphs
Recommendations
Metric properties of the Tower of Hanoi graphs and Stern's diatomic sequence
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. ...
Optimality of an algorithm solving the Bottleneck Tower of Hanoi problem
We study the Bottleneck Tower of Hanoi puzzle posed by D. Wood in 1981. There, a relaxed placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a pregiven value k. A shortest sequence of moves (...
Which multi-peg tower of hanoi problems are exponential?
WG'12: Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer ScienceConnectivity properties are very important characteristics of a graph. Whereas it is usually referred to as a measure of a graph's vulnerability, a relatively new approach discusses a graph's average connectivity as a measure for the graph's performance ...
Reviews
Bruce E. Litow
An extension of the well-known Towers of Hanoi puzzle-a fixture in introducing recursive algorithms in programming-is explained in this paper. The extension is to regard the pegs as vertices in a digraph, so that adjacency becomes a constraint. Since a complete characterization of digraphs for which the Hanoi puzzle is solvable is known, namely graphs whose transitive closure contains a 3-clique, the authors are interested in infinite graph families that admit optimal solutions. The solvability characterization is obvious: in such a graph, there is always a path to a 3-clique; if the pegs are the clique's vertices, then the situation reduces immediately to the classical Hanoi puzzle. The graph families studied are motivated directly by the observation on solvability. The authors define these families in a somewhat elliptical manner and give only diagrams. The bulk of the paper contains numerous inductive arguments and the main interest attaches to the optimality arguments. The move counts obtained are quite similar to the classical case, so nothing remarkable emerges on that account. I must admit that while these arguments may be of use in analyzing certain kinds of combinatorial routines, the paper is so specialized that it seems to fall outside the scope of this journal. Online Computing Reviews Service
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments
Information & Contributors
Information
Published In
Copyright © 2008 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 12 December 2008
Accepted: 01 July 2008
Revised: 01 June 2008
Received: 01 March 2006
Published in TALG Volume 5, Issue 1
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 377Total Downloads
- Downloads (Last 12 months)3
- Downloads (Last 6 weeks)1
Reflects downloads up to 02 Sep 2024
Other Metrics
Citations
Cited By
View all- Leiss E(2013)The worst Hanoi graphsTheoretical Computer Science10.1016/j.tcs.2013.05.020498(100-106)Online publication date: 1-Aug-2013
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in