The power of a pebble: exploring and mapping directed graphs

Exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts. Many results have focused on finding efficient solutions to restricted versions of the problem. In this paper, we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting. We model the environment by an unknown directed graph G, and consider the problem of a robot exploring and mapping G. The edges emanating from each vertex are numbered from `1? to `d?, but we do not assume that the vertices of G are labeled. Since the robot has no way of distinguishing between vertices, it has no hope of succeeding unless it is given some means of distinguishing between vertices. For this reason we provide the robot with a “pebble”?a device that it can place on a vertex and use to identify the vertex later. In this paper we show: (1) If the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble. (2) If the robot does not know an upper bound on the number of vertices n, then ?(log log n) pebbles are both necessary and sufficient. In both cases our algorithms are deterministic.

[1]  Kenneth Basye,et al.  Inferring Finite Automata with Stochastic Output Functions and an Application to Map Learning , 2004, Machine Learning.

[2]  Xiaotie Deng,et al.  How to learn an unknown environment. I: the rectilinear case , 1998, JACM.

[3]  Xiaotie Deng,et al.  Robot Map Verification of a Graph World , 1999, WADS.

[4]  Manuel Blum,et al.  On the power of the compass (or, why mazes are easier to search than graphs) , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[5]  Ronald L. Rivest,et al.  Diversity-based inference of finite automata , 1994, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[6]  Xiaotie Deng,et al.  Exploring an unknown graph , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[7]  Michael R. M. Jenkin,et al.  Map validation and robot self-location in a graph-like world , 1997, Robotics Auton. Syst..

[8]  Jeffery R. Westbrook,et al.  Robot navigation with range queries , 1996, STOC '96.

[9]  Dana Angluin,et al.  Learning Regular Sets from Queries and Counterexamples , 1987, Inf. Comput..

[10]  Michael Jenkin,et al.  Robotic exploration as graph construction , 1991, IEEE Trans. Robotics Autom..

[11]  Mona Singh,et al.  Piecemeal Graph Exploration by a Mobile Robot , 1999, Inf. Comput..

[12]  Avrim Blum,et al.  An Online Algorithm for Improving Performance in Navigation , 2000, SIAM J. Comput..

[13]  Edward A. Feigenbaum,et al.  Switching and Finite Automata Theory: Computer Science Series , 1990 .

[14]  Ricardo A. Baeza-Yates,et al.  Searching in the Plane , 1993, Inf. Comput..

[15]  Rolf Klein,et al.  A competitive strategy for learning a polygon , 1997, SODA '97.

[16]  Mona Singh,et al.  Piecemeal graph exploration by a mobile robot (extended abstract) , 1995, COLT '95.

[17]  Yoram Singer,et al.  A Comparison of New and Old Algorithms for a Mixture Estimation Problem , 2004, Machine Learning.

[18]  Leslie Pack Kaelbling,et al.  Uncertainty in Graph-Based Map Learning , 1993 .

[19]  Michael A. Bender,et al.  The power of team exploration: two robots can learn unlabeled directed graphs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[20]  Baruch Schieber,et al.  Navigating in Unfamiliar Geometric Terrain , 1997, SIAM J. Comput..

[21]  Xiaotie Deng,et al.  Competitive robot mapping with homogeneous markers , 1996, IEEE Trans. Robotics Autom..

[22]  Amos Fiat,et al.  Randomized robot navigation algorithms , 1996, SODA '96.

[23]  Amos Fiat,et al.  Online Navigation in a Room , 1992, J. Algorithms.

[24]  Ronitt Rubinfeld,et al.  Efficient Learning of Typical Finite Automata from Random Walks , 1997, Inf. Comput..

[25]  G. KILIBARDA,et al.  On behaviour of automata in labyrinths , 1993 .

[26]  Ronald L. Rivest,et al.  Inference of finite automata using homing sequences , 1989, STOC '89.

[27]  Avrim Blum,et al.  An on-line algorithm for improving performance in navigation , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[28]  Mona Singh,et al.  Piecemeal Learning of an Unknown Environment , 1993, COLT.

[29]  Andrzej Pelc,et al.  Exploring unknown undirected graphs , 1999, SODA '98.

[30]  Ronitt Rubinfeld,et al.  Exactly learning automata with small cover time , 1995, COLT '95.

[31]  Ronitt Rubinfeld,et al.  Efficient algorithms for learning to play repeated games against computationally bounded adversaries , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[32]  Susanne Albers,et al.  Exploring unknown environments , 1997, STOC '97.

[33]  Michael A. Bender,et al.  The power of a pebble: exploring and mapping directed graphs , 1998, STOC '98.

[34]  Anupam N. Shah.  Pebble automata on arrays , 1974, Comput. Graph. Image Process..

[35]  Mihalis Yannakakis,et al.  Shortest Paths Without a Map , 1989, Theor. Comput. Sci..

[36]  Elizabeth M. Belding-Royer,et al.  A review of current routing protocols for ad hoc mobile wireless networks , 1999, IEEE Wirel. Commun..