Euler tour lock-in problem in the rotor-router model: I choose pointers and you choose port numbers

The rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph G=(V,E), where |V|=n and |E|=m, adopted by an agent controlled by the rotor-router mechanism forms eventually 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 recent work [11] Yanovski et al. 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. The case study is performed in the form of a game between a player intending to lock-in the agent in an Euler tour as quickly as possible and its adversary with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time O(m). On the other hand we show that if adversary is solely responsible for the assignment of ports and pointers, the lock-in time Ω(m•D) can be enforced in any graph with m edges and diameter D. Furthermore, we show that if provides its own port numbering after the initial setup of pointers by , the complexity of the lock-in problem is bounded by O(m • min {logm,D}). We also propose a class of graphs in which the lock-in requires time Ω(m •logm). In the remaining two cases we show that the lock-in requires time Ω(m •D) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is O(m).