TY - JOUR

T1 - The power of a Pebble

T2 - Exploring and mapping directed graphs

AU - Bender, Michael A.

AU - Fernández, Antonio

AU - Ron, Dana

AU - Sahai, Amit

AU - Vadhan, Salil

N1 - Funding Information:
1A preliminary version of this work appeared in STOC ‘98 [9]. 2 This work was done while the author was at the Division of Engineering and Applied Sciences, Harvard University, and was supported by NSF Grants CCR-95-04436 and CCR-93-13775. 3 Supported by the Spanish Ministry of Education, Army Grant DAAH04-95-1-0607, and ARPA Contract N00014-95-1-1246. This work was done while the author was at the Laboratory for Computer Science, MIT. 4This work was done while the author was at the Laboratory for Computer Science, MIT, and was supported by an NSF postdoctoral grant and by an ONR Science Scholar Fellowhip at the Bunting Institute. 5,6Supported by a DOD NDSEG doctoral fellowship and partially by DARPA Grant DABT63-96-C-0018. 1

PY - 2002/7/10

Y1 - 2002/7/10

N2 - 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 Θ (log log n) pebbles are both necessary and sufficient. In both cases our algorithms are deterministic.

AB - 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 Θ (log log n) pebbles are both necessary and sufficient. In both cases our algorithms are deterministic.

UR - http://www.scopus.com/inward/record.url?scp=0037055377&partnerID=8YFLogxK

U2 - 10.1006/inco.2001.3081

DO - 10.1006/inco.2001.3081

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AN - SCOPUS:0037055377

SN - 0890-5401

VL - 176

SP - 1

EP - 21

JO - Information and Computation

JF - Information and Computation

IS - 1

ER -