Graph exploration by a finite automaton

  • Authors:

  • Pierre Fraigniaud;David Ilcinkas;Guy Peer;Andrzej Pelc;David Peleg

  • Affiliations:

  • CNRS, LRI, Université Paris-Sud, France;CNRS, LRI, Université Paris-Sud, France;Department of Computer Science, Weizmann Institute, Israel;Dépt. of d'informatique, Univ. du Québec en Outaouais, Canada;Department of Computer Science, Weizmann Institute, Israel

  • Venue:
  • Theoretical Computer Science - Mathematical foundations of computer science 2004
  • Year:
  • 2005

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Abstract

A finite automaton, simply referred to as a robot, has to explore a graph whose nodes are unlabeled and whose edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the graph or of its size. Its task is to traverse all the edges of the graph. We first show that, for any K-state robot and any d ≥ 3, there exists a planar graph of maximum degree d with at most K + 1 nodes that the robot cannot explore. This bound improves all previous bounds in the literature. More interestingly, we show that, in order to explore all graphs of diameter D and maximum degree d, a robot needs Ω(D log d) memory bits, even if we restrict the exploration to planar graphs. This latter bound is tight. Indeed, a simple DFS up to depth D + 1 enables a robot to explore any graph of diameter D and maximum degree d using a memory of size O(D log d) bits. We thus prove that the worst case space complexity of graph exploration is Θ(D log d) bits.