# Shortest paths in the Tower of Hanoi graph and finite automata

@article{Romik2006ShortestPI,
title={Shortest paths in the Tower of Hanoi graph and finite automata},
author={Dan Romik},
journal={SIAM J. Discret. Math.},
year={2006},
volume={20},
pages={610-622}
}
• D. Romik
• Published 2006
• Mathematics, Computer Science
• SIAM J. Discret. Math.
We present efficient algorithms for constructing a shortest path between two configurations in the Tower of Hanoi graph and for computing the length of the shortest path. The key element is a finite-state machine which decides, after examining on the average only a small number of the largest discs (asymptotically, $\frac{63}{38} \approx 1.66$), whether the largest disc will be moved once or twice. This solves a problem raised by Andreas Hinz and results in a better understanding of how the… Expand
71 Citations
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#### References

SHOWING 1-10 OF 18 REFERENCES
Metric properties of the Tower of Hanoi graphs and Stern's diatomic sequence
• Computer Science, Mathematics
• Eur. J. Comb.
• 2005
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 in the Tower of Hanoi graphs, and implies that only for vertices of degree two this number is zero. Expand
An analysis of the generalized Towers of Hanoi problem
• M. Er
• Computer Science, Mathematics
• BIT
• 1983
The shortest-path tree clearly characterizes the generalized Towers of Hanoi problem; and its use leads to a very simple analysis of the generalized problem. Expand
Shortest paths between regular states of the Tower of Hanoi
• A. Hinz
• Mathematics, Computer Science
• Inf. Sci.
• 1992
Two generalizations of the classical Tower of Hanoi problem 0, i.e., to transfer n discs from one peg to another by a series of legal moves, are considered and some mathematical background is developed to construct simple algorithms that solve problems 1 and 2 correctly and return the minimum number of moves. Expand
Graphs S(n, k) and a Variant of the Tower of Hanoi Problem
• Mathematics
• 1997
For any n ≥ 1 and any k ≥ 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs areExpand
A statistical analysis of the towers of hanoi problem
The Towers of Hanoi problem is analyzed with the aid of Hanoi graphs depicting legal configurations (i. e. arrangements of the disks on the pegs) and the transitions among them. Specifically, theExpand
The average distance on the Sierpiński gasket
• Mathematics
• 1990
SummaryThe canonical distance of points on the Sierpiński gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysisExpand
On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem
• Computer Science, Mathematics
• Discret. Appl. Math.
• 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.
The Tower of Hanoi: A historical survey and bibliography
• version 0.2. Preprint
• 2001
The Tower of Hanoi. Algebras and Combinatorics (ICAC'97, Hong Kong)
• The Tower of Hanoi. Algebras and Combinatorics (ICAC'97, Hong Kong)
• 1999
Graphs S(n
• k) and a variant of the Tower of Hanoi problem. Czechoslovak Math. J. 47 (122), 95–104
• 1997