Robustness of the Rotor–Router Mechanism

Abstract

The rotor–router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer $\pi _v$ which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the “next exit port”). The rotor–router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor–router mechanism eventually forms an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In Yanovski et al. (Algorithmica 37(3):165–186, 2003), it was proved that, independently of the initial configuration of the rotor–router mechanism in G, the agent locks-in in time bounded by $2mD$, where $D$ is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor–router mechanism. Our analysis is performed in the form of a game between a player ${\mathcal {P}}$ intending to lock-in the agent in an Euler tour as quickly as possible and its adversary ${\mathcal {A}}$ with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values $\pi _v$. We show, for example, that if ${\mathcal {A}}$ provides its own port numbering after the initial setup of pointers by ${\mathcal {P}}$, the worst-case complexity of the lock-in problem is ${\varTheta }(m\cdot \min \{\log m,D\})$. We also investigate the robustness of the rotor–router graph exploration in presence of faults in the pointers $\pi _v$ or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within $\mathcal {O}(km)$ steps.