9 research outputs found
Robustness of the Rotor-Router Mechanism
International audienceThe rotor-router model, also called the Propp machine, was first considered as a deter-ministic 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 π(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 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 [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 P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values π(v). We show, for example, that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is O(m·min{log m, D}). We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers π(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 O(km) steps
Lock-in Problem for Parallel Rotor-router Walks
The rotor-router model, also called the Propp machine, was introduced as a
deterministic alternative to the random walk. In this model, a group of
identical tokens are initially placed at nodes of the graph. Each node
maintains a cyclic ordering of the outgoing arcs, and during consecutive turns
the tokens are propagated along arcs chosen according to this ordering in
round-robin fashion. The behavior of the model is fully deterministic. Yanovski
et al.(2003) proved that a single rotor-router walk on any graph with m edges
and diameter stabilizes to a traversal of an Eulerian circuit on the set of
all 2m directed arcs on the edge set of the graph, and that such periodic
behaviour of the system is achieved after an initial transient phase of at most
2mD steps. The case of multiple parallel rotor-routers was studied
experimentally, leading Yanovski et al. to the conjecture that a system of k
\textgreater{} 1 parallel walks also stabilizes with a period of length at
most steps. In this work we disprove this conjecture, showing that the
period of parallel rotor-router walks can in fact, be superpolynomial in the
size of graph. On the positive side, we provide a characterization of the
periodic behavior of parallel router walks, in terms of a structural property
of stable states called a subcycle decomposition. This property provides us the
tools to efficiently detect whether a given system configuration corresponds to
the transient or to the limit behavior of the system. Moreover, we provide
polynomial upper bounds of and on the
number of steps it takes for the system to stabilize. Thus, we are able to
predict any future behavior of the system using an algorithm that takes
polynomial time and space. In addition, we show that there exists a separation
between the stabilization time of the single-walk and multiple-walk
rotor-router systems, and that for some graphs the latter can be asymptotically
larger even for the case of walks
The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks
International audienceThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work Research partially supported by the ANR Project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of the Investments for the future Programme IdEx Bordeaux-CP
Robustness of the Rotor-router Mechanism
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