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 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 P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary 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 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 A provides its own port numbering after the initial setup of pointers by 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).

Derandomizing random walks in undirected graphs using locally fair exploration strategies

“…Consequently, Oldest-First explorations on symmetric directed graphs are fair, in the sense that all edges are visited with the same frequency f e (G) = 1/(2|E|). Other studies of the rotor-router model include specific graph classes [13], adversarial scenarios [4], and aspects of fault-tolerance [5].…”

Bounds on the cover time of parallel rotor walks

“…As a starting point, the proof uses a decomposition of the edge set of a graph, introduced by Bampas et al [3], into a "heavy part" containing a constant proportion of the edges and a "deep part", having diameter linear in D. The main part of the analysis is to show that an appropriate initialization of k agents in the heavy part takes a long time to reach the most distant nodes of the deep part. The argument also takes advantage of the delayed deployment technique.…”

“…In such scenario, the exploration will be completed after time at least D > mD k . Thus, we can safely assume that k ≤ m. For any graph G = (V, E), as shown in [3,Theorem 2], there exists a partition of the edge set E = E 1 ∪ E 2 , such that (see Fig. 2 for an illustration):…”

“…In general, it is known that for any n-node graph of m edges and diameter D, the cover time of the rotor-router in a worst-case initialization is precisely Θ(mD) [16,3]. By comparison, the random walk satisfies an upper bound of O(mD log n) on the cover time.…”

Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

“…and N − (v) respectively denote the outneighborhood and the in-neighborhood 3 and aperiodic 4 . It is well known that any ergodic P has a unique stationary distribution π ∈ R n >0 (i.e., πP = π), and the limit distribution is π (i.e., lim t→∞ ξP t = π for any probability distribution ξ ∈ R n ≥0 on V ).…”