Distributed Algorithms for Unidirectional Networks
SIAM Journal on Computing
Simulating a Random Walk with Constant Error
Combinatorics, Probability and Computing
Deterministic random walks on the two-dimensional grid
Combinatorics, Probability and Computing
The cover time of deterministic random walks
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
Boundary patrolling by mobile agents with distinct maximal speeds
ESA'11 Proceedings of the 19th European conference on Algorithms
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks
Proceedings of the 2013 ACM symposium on Principles of distributed computing
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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(mċD) 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(mċmin{log m, D}). We also propose a class of graphs in which the lock-in requires time Ω(m ċ log m). In the remaining two cases we show that the lock-in requires time Ω(m ċ 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).