Robustness of the Rotor‐Router Mechanism

Robustness of the Rotor‐Router Mechanism

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Article ID: iaor20173015
Volume: 78
Issue: 3
Start Page Number: 869
End Page Number: 895
Publication Date: Jul 2017
Journal: Algorithmica
Authors: , , , , , ,
Keywords: combinatorial optimization, optimization, graphs
Abstract:

The rotor–router model, also called the Propp machine, was first considered as a deterministic 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 equ1 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 eventually forms 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 2 m D equ2 , where D equ3 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 equ4 intending to lock‐in the agent in an Euler tour as quickly as possible and its adversary A equ5 with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values π v equ6 . We show, for example, that if A equ7 provides its own port numbering after the initial setup of pointers by P equ8 , the worst‐case complexity of the lock‐in problem is Θ ( m · min { log m , D } ) equ9 . We also investigate the robustness of the rotor–router graph exploration in presence of faults in the pointers π v equ10 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 ( k m ) equ11 steps.

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