2009, English, Article edition: Euler Tour Lock-in Problem in the Rotor-Router Model Bampas, Evangelos; Gasieniec, Leszek; Hanusse, Nicolas; ...

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Bookmark: http://trove.nla.gov.au/version/201161
Physical Description
  • text
Published
  • HAL - CCSD
  • 2009
Language
  • English

Edition details

Title
  • Euler Tour Lock-in Problem in the Rotor-Router Model
Author
  • Bampas, Evangelos
  • Gasieniec, Leszek
  • Hanusse, Nicolas
  • Ilcinkas, David
  • Klasing, Ralf
  • Kosowski, Adrian
Published
  • HAL - CCSD
  • 2009
Physical Description
  • text
Subjects
Notes
  • 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 [Yan03] 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 2m diam, where diam is the diameter of G. This upper bound can be matched asymptotically in lollipop graphs. 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 pl intending to lock-in the agent in an Euler tour as quickly as possible and its adversary ad 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 ad is solely responsible for the assignment of ports and pointers, the lock-in time Omega(m diam) can be enforced in any graph with m edges and diameter diam. Furthermore, we show that if ad provides its own port numbering after the initial setup of pointers by pl, the complexity of the lock-in problem is bounded by O(m min{log m,diam}). We also propose a class of graphs in which the lock-in requires time Omega(m log m). In the remaining two cases we show that the lock-in requires time Omega(m diam) 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).
  • HAL:hal-00402749, version 1
  • HAL:http:/​/​hal.archives-ouvertes.fr/​hal-00402749/​en/​
Language
  • English
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