Robustness of the Rotor-router Mechanism

• Authors: Evangelos Bampas;Leszek Gąsieniec;Ralf Klasing;Adrian Kosowski;Tomasz Radzik
• Venue: OPODIS '09 Proceedings of the 13th International Conference on Principles of Distributed Systems
• Year: 2009

We consider the model of exploration of an undirected graph G by a single agent which is called the rotor-router mechanism or the Propp machine (among other names). Let *** v indicate the edge adjacent to a node v which the agent took on its last exit from v . The next time when the agent enters node v , first a "rotor" at node v advances pointer *** v to the edge ${\it next}(\pi_v)$ which is next after the edge *** v in a fixed cyclic order of the edges adjacent to v . Then the agent is directed onto edge *** v to move to the next node. It was shown before that after initial O (mD ) steps, the agent periodically follows one established Eulerian cycle, that is, in each period of 2m consecutive steps the agent traverses each edge exactly twice, once in each direction. The parameters m and D are the number of edges in G and the diameter of G . We investigate robustness of such exploration in presence of faults in the pointers *** v or dynamic changes in the graph. We show that after the exploration establishes an Eulerian cycle, if at some step the values of k pointers *** v are arbitrarily changed, then a new Eulerian cycle is established within O (km ) steps; if at some step k edges are added to the graph, then a new Eulerian cycle is established within O (km ) steps; if at some step an edge is deleted from the graph, then a new Eulerian cycle is established within O (***m ) steps, where *** is the smallest number of edges in a cycle in graph G containing the deleted edge. Our proofs are based on the relation between Eulerian cycles and spanning trees known as the "BEST" Theorem (after de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte).