Optimal constrained graph exploration

Abstract

We address the problem of exploring an unknown graph G = (V, E) from a given start node s 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 l, then the robot must remain within distance l from the start node s. In the second variation, a fuel tank of limited capacity forces the robot to return to s after traversing C 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 O(¦E¦) 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 O(¦E¦) algorithm. This improves on the previous best known bound of O(¦E¦ + ¦V¦log 2¦V¦) in [4]. Since the lower bound for the graph exploration problems is ¦E¦, our algorithm is optimal, thus answering the open problem of Awerbuch, Betke, Rivest, and Singh [3].