# Euler Tour Lock-In Problem in the Rotor-Router Model

## Abstract

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 2*mD*, 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 \(\cal P\) intending to lock-in the agent in an Euler tour as quickly as possible and its adversary \(\cal A\) 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 \(\cal A\) 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 \(\cal A\) provides its own port numbering after the initial setup of pointers by \(\cal P\), the complexity of the lock-in problem is bounded by *O*(*m* · min {log*m*,*D*}). We also propose a class of graphs in which the lock-in requires time Ω(*m* ·log*m*). 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*).

## Keywords

Random Walk Undirected Graph Edge Incident Complete Bipartite Graph Port Number## Preview

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