Robustness of the Rotor-router Mechanism

Abstract

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 π _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 π _v to the edge ${\it next}(\pi_v)$ which is next after the edge π _v in a fixed cyclic order of the edges adjacent to v. Then the agent is directed onto edge π _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 π _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 π _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(γm) steps, where γ 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).