An efficient algorithm to determine all shortest paths in Sierpiński graphs

In the Switching Tower of Hanoi interpretation of Sierpiński graphsSpn, the P2 decision problem is to find out whether the largest moving disc has to be transferred once or twice in a shortest path between two given states/vertices. We construct an essentially optimal algorithm thus extending Romik’...

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Bibliographic Details
Published inDiscrete Applied Mathematics Vol. 177; pp. 111 - 120
Main Authors Hinz, Andreas M., Holz auf der Heide, Caroline
Format Journal Article
LanguageEnglish
Published Elsevier B.V 20.11.2014
Subjects
Automata
Shortest path algorithm
Sierpiński graphs
Switching Tower of Hanoi
Sierpiński graphs
Switching Tower of Hanoi
Shortest path algorithm
Automata
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Summary:In the Switching Tower of Hanoi interpretation of Sierpiński graphsSpn, the P2 decision problem is to find out whether the largest moving disc has to be transferred once or twice in a shortest path between two given states/vertices. We construct an essentially optimal algorithm thus extending Romik’s approach for p=3 to the general case. The algorithm makes use of three automata and the underlying theory includes a simple argument for the fact that there are at most two shortest paths between any two vertices. The total number of pairs leading to non-unique solutions is determined and employing a Markov chain argument it is shown that the number of input pairs needed for the decision is bounded above by a number independent of n. Elementary algorithms for the length of the shortest path(s) and the best first move/edge are also presented.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2014.05.049