Finding short right-hand-on-the-wall walks in graphs

  • Authors:

  • Stefan Dobrev;Jesper Jansson;Kunihiko Sadakane;Wing-Kin Sung

  • Affiliations:

  • SITE, University of Ottawa, Canada;Department of Computer Science and Information Systems, The University of Hong Kong, Hong Kong;Department of Computer Science and Communication Engineering, Kyushu University, Japan;School of Computing, National University of Singapore

  • Venue:
  • SIROCCO'05 Proceedings of the 12th international conference on Structural Information and Communication Complexity
  • Year:
  • 2005

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Abstract

We consider the problem of perpetual traversal by a single agent in an anonymous undirected graph G. Our requirements are: (1) deterministic algorithm, (2) each node is visited within O(n) moves, (3) the agent uses no memory, it can use only the label of the link via which it arrived to the current node, (4) no marking of the underlying graph is allowed and (5) no additional information is stored in the graph (e.g. routing tables, spanning tree) except the ability to distinguish between the incident edges (called Local Orientation). This problem is unsolvable, as has been proven in [9,28] even for much less restrictive setting. Our approach is to somewhat relax the requirement (5). We fix the following traversal algorithm: “Start by taking the edge with the smallest labelx. Afterwards, whenever you come to a node, continue by taking the successor edge (in the local orientation) to the edge via which you arrived” and ask whether it is for every undirected graph possible to assign the local orientations in such a way that the resulting perpetual traversal visits every node in O(n) moves. We give a positive answer to this question, by showing how to construct such local orientations. This leads to an extremely simple, memoryless, yet efficient traversal algorithm.