Optimal constrained graph exploration

We address the problem of exploring an unknown graph <i>G</i> = (<i>V</i>, <i>E</i>) from a given start node <i>s</i> with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges of the graph, i.e, it cannot jump between non-adjacent vertices. In the tethered robot case, if the tether (rope) has length <i>l</i>, then the robot must remain within distance <i>l</i> from the start node <i>s</i>. In the second variation, a fuel tank of limited capacity forces the robot to return to <i>s</i> after traversing <i>C</i> edges. The efficiency of algorithms for both variations of the problem is measured by the number of edges traversed during the exploration. We present an algorithm for a tethered robot which explores the graph in <i>&Ogr;</i>(¦<i>E</i>¦) edge traversals. The problem of exploration using a robot with a limited fuel tank capacity can be solved with a simple reduction from the tethered robot case and also yields a <i>&Ogr;</i>(¦<i>E</i>¦) algorithm. This improves on the previous best known bound of <i>&Ogr;</i>(¦<i>E</i>¦ + ¦<i>V</i>¦log <sup>2</sup>¦<i>V</i>¦) in [4]. Since the lower bound for the graph exploration problems is ¦<i>E</i>¦, our algorithm is optimal, thus answering the open problem of Awerbuch, Betke, Rivest, and Singh [3].

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