TY - JOUR
T1 - Fast periodic graph exploration with constant memory
AU - Gasieniec, Leszek
AU - Klasing, Ralf
AU - Martin, Russell
AU - Navarra, Alfredo
AU - Zhang, Xiaohui
N1 - Funding Information:
✩ Preliminary results concerning this paper appeared in [L. Gąsieniec, R. Klasing, R. Martin, A. Navarra, X. Zhang, Fast periodic graph exploration with constant memory, in: Proc. 14th Colloquium on Structural Information and Communication Complexity, SIROCCO 2007, in: Lecture Notes in Comput. Sci., vol. 4474, Springer-Verlag, 2007, pp. 26–40. [18]]. ✩✩ The 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 utilisation of the Internet and other distributed systems,” and by a visiting fellowship from LaBRI/ENSEIRB. * Corresponding author. E-mail addresses: leszek@csc.liv.ac.uk (L. Gąsieniec), ralf.klasing@labri.fr (R. Klasing), martin@csc.liv.ac.uk (R. Martin), navarra@dipmat.unipg.it (A. Navarra), cloud@csc.liv.ac.uk (X. Zhang). 1 The work was done while the author was a Post-Doc at LaBRI, France.
PY - 2008/8/1
Y1 - 2008/8/1
N2 - 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 coloured. 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. In this paper, we are interested in minimisation of the length of the exploration period. In other words, we want to minimise the maximum number of edge traversals performed by the robot between two consecutive visits of a generic node, in the same state and entering the node by the same port. Note that the problem is unsolvable if the local port numbers are set arbitrarily, see [L. Budach, Automata and labyrinths, Math. Nachr. 86 (1978) 195-282]. 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 [S. Dobrev, J. Jansson, K. Sadakane, W.-K. Sung, Finding short right-hand-on-the-wall walks in graphs, in: Proc. 12th Colloquium on Structural Information and Communication Complexity, SIROCCO 2005, in: Lecture Notes in Comput. Sci., vol. 3499, Springer, Berlin, 2005, pp. 127-139] that for oblivious robots π (n) ≤ 10 n. Recently Ilcinkas proposed another port labelling algorithm for robots equipped with two extra memory bits, see [D. Ilcinkas, Setting port numbers for fast graph exploration, in: Proc. 13th Colloquium on Structural Information and Communication Complexity, SIROCCO 2006, in: Lecture Notes in Comput. Sci., vol. 4056, Springer, Berlin, 2006, pp. 59-69], where the exploration period π (n) ≤ 4 n - 2. In the same paper, it is conjectured that the bound 4 n - 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) < 3.75 n - 2 steps hence decreasing the existing upper bound. This reduces the gap with the lower bound of π (n) ≥ 2 n - 2 holding for any robot.
AB - 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 coloured. 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. In this paper, we are interested in minimisation of the length of the exploration period. In other words, we want to minimise the maximum number of edge traversals performed by the robot between two consecutive visits of a generic node, in the same state and entering the node by the same port. Note that the problem is unsolvable if the local port numbers are set arbitrarily, see [L. Budach, Automata and labyrinths, Math. Nachr. 86 (1978) 195-282]. 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 [S. Dobrev, J. Jansson, K. Sadakane, W.-K. Sung, Finding short right-hand-on-the-wall walks in graphs, in: Proc. 12th Colloquium on Structural Information and Communication Complexity, SIROCCO 2005, in: Lecture Notes in Comput. Sci., vol. 3499, Springer, Berlin, 2005, pp. 127-139] that for oblivious robots π (n) ≤ 10 n. Recently Ilcinkas proposed another port labelling algorithm for robots equipped with two extra memory bits, see [D. Ilcinkas, Setting port numbers for fast graph exploration, in: Proc. 13th Colloquium on Structural Information and Communication Complexity, SIROCCO 2006, in: Lecture Notes in Comput. Sci., vol. 4056, Springer, Berlin, 2006, pp. 59-69], where the exploration period π (n) ≤ 4 n - 2. In the same paper, it is conjectured that the bound 4 n - 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) < 3.75 n - 2 steps hence decreasing the existing upper bound. This reduces the gap with the lower bound of π (n) ≥ 2 n - 2 holding for any robot.
KW - Finite automaton
KW - Full degree
KW - Penalty
KW - Period
KW - Port number
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U2 - 10.1016/j.jcss.2007.09.004
DO - 10.1016/j.jcss.2007.09.004
M3 - Article
AN - SCOPUS:43649102091
SN - 0022-0000
VL - 74
SP - 808
EP - 822
JO - Journal of Computer and System Sciences
JF - Journal of Computer and System Sciences
IS - 5
T2 - 14th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2007
Y2 - 5 June 2007 through 8 June 2007
ER -