The power of a Pebble: Exploring and mapping directed graphs

Michael A. Bender*, Antonio Fernández, Dana Ron, Amit Sahai, Salil Vadhan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

85 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalInformation and Computation
Volume176
Issue number1
DOIs
StatePublished - 10 Jul 2002

Funding

FundersFunder number
Bunting Institute
Division of Engineering and Applied Sciences
Laboratory for Computer Science
Spanish Ministry of Education, ArmyDAAH04-95-1-0607
National Science FoundationCCR-93-13775, CCR-95-04436
U.S. Department of Defense
Office of Naval Research
Defense Advanced Research Projects AgencyDABT63-96-C-0018
Massachusetts Institute of Technology
Harvard University
Advanced Research Projects AgencyN00014-95-1-1246

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