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Derandomizing random walks in undirected graphs using locally fair exploration strategies

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Abstract

We consider the problem of exploring an anonymous undirected graph using an oblivious robot. The studied exploration strategies are designed so that the next edge in the robot’s walk is chosen using only local information, and so that some local equity (fairness) criterion is satisfied for the adjacent undirected edges. Such strategies can be seen as an attempt to derandomize random walks, and are natural counterparts for undirected graphs of the rotor-router model for symmetric directed graphs. The first of the studied strategies, known as Oldest-First, always chooses the neighboring edge for which the most time has elapsed since its last traversal. Unlike in the case of symmetric directed graphs, we show that such a strategy in some cases leads to exponential cover time. We then consider another strategy called Least-Used-First which always uses adjacent edges which have been traversed the smallest number of times. We show that any Least-Used-First exploration covers a graph G = (V, E) of diameter D within time O(D|E|), and in the long run traverses all edges of G with the same frequency.

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Authors and Affiliations

  1. Department of Computer Science, King’s College London, London, WC2R 2LS, UK

    Colin Cooper

  2. LaBRI - Université de Bordeaux - CNRS, 351 cours de la Libération, 33405, Talence, France

    David Ilcinkas & Ralf Klasing

  3. INRIA Bordeaux Sud-Ouest, 351 cours de la Libération, 33405, Talence, France

    Adrian Kosowski

Authors
  1. Colin Cooper
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  2. David Ilcinkas
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  3. Ralf Klasing
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  4. Adrian Kosowski
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Corresponding author

Correspondence to Ralf Klasing.

Additional information

The research was partially supported by the ANR project “ALADDIN”, the INRIA équipe-project “CEPAGE”, and the Royal Society IJP grant “Random walks, interacting particles and faster network exploration”.

A. Kosowski was partially supported by Polish Ministry Grant N206 491738.

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Cooper, C., Ilcinkas, D., Klasing, R. et al. Derandomizing random walks in undirected graphs using locally fair exploration strategies. Distrib. Comput. 24, 91–99 (2011). https://doi.org/10.1007/s00446-011-0138-4

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  • DOI: https://doi.org/10.1007/s00446-011-0138-4

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