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Graph Decomposition for Memoryless Periodic Exploration

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Abstract

We consider a general framework in which a memoryless robot periodically explores all the nodes of a connected anonymous graph by following local information available at each vertex. For each vertex v, the endpoints of all edges adjacent to v are assigned unique labels within the range 1 to deg (v) (the degree of v). The generic exploration strategy is implemented using a right-hand-rule transition function: after entering vertex v via the edge labeled i, the robot proceeds with its exploration, leaving via the edge having label [i mod deg (v)]+1 at v.

A lot of attention has been given to the problem of labeling the graph so as to achieve a periodic exploration having the minimum possible length π. It has recently been proved (Czyzowicz et al., Proc. SIROCCO’09, 2009) that \(\pi\leq4\frac{1}{3}n\) holds for all graphs of n vertices. Herein, we provide a new labeling scheme which leads to shorter exploration cycles, improving the general bound to π≤4n−2. This main result is shown to be tight with respect to the class of labellings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest.

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Author information

Correspondence to Alfredo Navarra.

Additional information

The research was partially funded by the State Committee for Scientific Research (Poland) Grant 4 T11C 047 25, by the ANR-project “ALADDIN” (France), and by the project “CEPAGE” of INRIA (France).

An extended abstract of this paper appeared in the proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science (MFCS) [12].

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Kosowski, A., Navarra, A. Graph Decomposition for Memoryless Periodic Exploration. Algorithmica 63, 26–38 (2012). https://doi.org/10.1007/s00453-011-9518-1

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Keywords

  • Anonymous graph
  • Labeling
  • Exploration algorithm
  • Oblivious robots