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 from 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 [1]] that \(\pi \leq 4\frac13 n\) 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 ≤ 4 n − 2. This main result is shown to be tight with respect to the class of labelings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest.
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).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Czyzowicz, J., Dobrev, S., Gąsieniec, L., Ilcinkas, D., Jansson, J., Klasing, R., Lignos, Y., Martin, R., Sadakane, K., Sung, W.: More efficient periodic traversal in anonymous undirected graphs. In: Proc. 16th Colloquium on Structural Information and Communication Complexity (SIROCCO). LNCS. Springer, Heidelberg (to appear, 2009)
Gąsieniec, L., Radzik, T.: Memory efficient anonymous graph exploration. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 14–29. Springer, Heidelberg (2008)
Dobrev, S., Jansson, J., Sadakane, K., Sung, W.-K.: Finding short right-hand-on-the-wall walks in graphs. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 127–139. Springer, Heidelberg (2005)
Budach, L.: Automata and labyrinths. Mathematische Nachrichten 86, 195–282 (1978)
Rollik, H.: Automaten in planaren graphen. Acta Informatica 13, 287–298 (1980)
Cook, S., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing 9(3), 636–652 (1980)
Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoretical Computer Science 345(2-3), 331–344 (2005)
Reingold, O.: Undirected st-connectivity in log-space. In: Proc. 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 376–385 (2005)
Fraigniaud, P., Ilcinkas, D., Pelc, A.: Impact of memory size on graph exploration capability. Discrete Applied Mathematics 156(12), 2310–2319 (2008)
Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Transactions on Algorithms 4(4), 1–18 (2008)
Ilcinkas, D.: Setting port numbers for fast graph exploration. Theoretical Computer Science 401(1-3), 236–242 (2008)
Gąsieniec, L., Klasing, R., Martin, R., Navarra, A., Zhang, X.: Fast periodic graph exploration with constant memory. Journal of Computer and System Sciences 74(5), 802–822 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kosowski, A., Navarra, A. (2009). Graph Decomposition for Improving Memoryless Periodic Exploration. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_43
Download citation
DOI: https://doi.org/10.1007/978-3-642-03816-7_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03815-0
Online ISBN: 978-3-642-03816-7
eBook Packages: Computer ScienceComputer Science (R0)