Abstract
We consider the problem of periodic exploration of all nodes in undirected graphs by using a finite state automaton called later a robot. The robot, using a constant number of states (memory bits), must be able to explore any unknown anonymous graph. The nodes in the graph are neither labelled nor colored. However, while visiting a node v the robot can distinguish between edges incident to it. The edges are ordered and labelled by consecutive integers 1,...,d(v) called port numbers, where d(v) is the degree of v. Periodic graph exploration requires that the automaton has to visit every node infinitely many times in a periodic manner. Note that the problem is unsolvable if the local port numbers are set arbitrarily, see [8]. In this context, we are looking for the minimum function π(n), such that, there exists an efficient deterministic algorithm for setting the local port numbers allowing the robot to explore all graphs of size n along a traversal route with the period π(n). Dobrev et al. proved in [13] that for oblivious robots π(n) ≤ 10n. Recently Ilcinkas proposed another port labelling algorithm for robots equipped with two extra memory bits, see [20], where the exploration period π(n) ≤ 4n − 2. In the same paper, it is conjectured that the bound 4n − O(1) is tight even if the use of larger memory is allowed. In this paper, we disprove this conjecture presenting an efficient deterministic algorithm arranging the port numbers, such that, the robot equipped with a constant number of bits is able to complete the traversal period in π(n) ≤ 3.75n − 2 steps hence decreasing the existing upper bound. This reduces the gap with the lower bound of π(n) ≥ 2n − 2 holding for any robot.
This research was partially funded by the project “ALPAGE” of the ANR “Masse de données: Modélisation, Simulation, Applications”, the project “CEPAGE” of INRIA, the European projects COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL), COST Action 295, “Dynamic Communication Networks” (DYNAMO), the Nuffield Foundation grant NAL/32566, “The structure and efficient utilization of the Internet and other distributed systems”, and by a visiting fellowship from LaBRI/ENSEIRB.
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References
Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM Journal on Computing 29, 1164–1188 (2000)
Awerbuch, B., et al.: Piecemeal Graph Exploration by a Mobile Robot. Information and Computation 152(2), 155–172 (1999)
Awerbuch, B., Kobourov, S.G.: Polylogarithmic-Overhead Piecemeal Graph Exploration. In: Proc. 11th Annual Conference on Computational Learning Theory (COLT 1998), pp. 280–286 (1998)
Bender, M.A., et al.: The Power of a Pebble: Exploring and Mapping Directed Graphs. Information and Computation 176(1), 1–21 (2002)
Bender, M.A., Slonim, D.: The power of team exploration: Two robots can learn unlabeled directed graphs. In: Proc. 35th Ann. Symp. on Foundations of Computer Science (FOCS 1994), pp. 75–85 (1994)
Betke, M., Rivest, R., Singh, M.: Piecemeal learning of an unknown environment. Machine Learning 18, 231–254 (1995)
Bhatia, R., et al.: The Full Degree Spanning Tree Problem. In: Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pp. 864–865 (1999)
Budach, L.: Automata and labyrinths. Math. Nachrichten 86, 195–282 (1978)
Cohen, R., et al.: Label-guided graph exploration by a finite automaton. In: Caires, L., et al. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 335–346. Springer, Heidelberg (2005)
Cook, S., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM J. on Computing 9(3), 636–652 (1980)
Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. Journal of Graph Theory 32, 265–297 (1999)
Diks, K., et al.: Tree exploration with little memory. Journal of Algorithms 51, 38–63 (2004)
Dobrev, S., et al.: 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)
Duncan, C.A., Kobourov, S.G., Kumar, V.S.A.: Optimal constrained graph exploration. In: 12th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA 2001), pp. 807–814. ACM Press, New York (2001)
Fleischer, R., Trippen, G.: Exploring an Unknown Graph Efficiently. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 11–22. Springer, Heidelberg (2005)
Fraigniaud, P., Ilcinkas, D.: Digraphs exploration with little memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004)
Fraigniaud, P., et al.: Graph exploration by a finite automaton. Theoretical Computer Science 345, 331–344 (2005)
Fraigniaud, P., et al.: The Reduced Automata Technique for Graph Exploration Space Lower Bounds, Essays in Memory of Shimon Even. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 1–26. Springer, Heidelberg (2006)
Gąsieniec, L., et al.: Tree exploration with logarithmic memory. In: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007) (2007)
Ilcinkas, D.: Setting port numbers for fast graph exploration. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 59–69. Springer, Heidelberg (2006)
Panaite, P., Pelc, A.: Exploring unknown undirected graphs. Journal of Algorithms 33, 281–295 (1999)
Reingold, O.: Undirected ST-Connectivity in Log-Space. In: Proc. 37th ACM Symposium on Theory of Computing (STOC 2005), pp. 376–385 (2005)
Rollik, H.: Automaten in planaren Graphen. Acta Informatica 13, 287–298 (1980)
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Gąsieniec, L., Klasing, R., Martin, R., Navarra, A., Zhang, X. (2007). Fast Periodic Graph Exploration with Constant Memory. In: Prencipe, G., Zaks, S. (eds) Structural Information and Communication Complexity. SIROCCO 2007. Lecture Notes in Computer Science, vol 4474. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72951-8_4
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