Euler tour lock-in problem in the rotor-router model: I choose pointers and you choose port numbers

Evangelos Bampas, Leszek Ga̧sieniec, Nicolas Hanusse, David Ilcinkas, Ralf Klasing, Adrian Kosowski

Research output: Chapter in Book/Report/Conference proceedingConference contribution

32 Scopus citations

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 intending to lock-in the agent in an Euler tour as quickly as possible and its adversary 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 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 provides its own port numbering after the initial setup of pointers by , the complexity of the lock-in problem is bounded by O(m • min {logm,D}). We also propose a class of graphs in which the lock-in requires time Ω(m •logm). 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).

Original languageEnglish (US)
Title of host publicationDistributed Computing - 23rd International Symposium, DISC 2009, Proceedings
Pages423-435
Number of pages13
DOIs
StatePublished - Dec 1 2009
Event23rd International Symposium on Distributed Computing, DISC 2009 - Elche, Spain
Duration: Sep 23 2009Sep 25 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5805 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference23rd International Symposium on Distributed Computing, DISC 2009
CountrySpain
CityElche
Period9/23/099/25/09

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Bampas, E., Ga̧sieniec, L., Hanusse, N., Ilcinkas, D., Klasing, R., & Kosowski, A. (2009). Euler tour lock-in problem in the rotor-router model: I choose pointers and you choose port numbers. In Distributed Computing - 23rd International Symposium, DISC 2009, Proceedings (pp. 423-435). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5805 LNCS). https://doi.org/10.1007/978-3-642-04355-0_44