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Derandomizing Random Walks in Undirected Graphs Using Locally Fair Exploration Strategies

  • Colin Cooper
  • David Ilcinkas
  • Ralf Klasing
  • Adrian Kosowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5556)

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 undirected counterparts of the rotor-router model for symmetric directed graphs.

The first of the studied strategies, known as Oldest-First (OF), 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 (LUF) 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 \(\mathit{D}\) within time \(O(\mathit{D}|E|)\), and in the long run traverses all edges of G with the same frequency.

Keywords

Random Walk Undirected Graph Time Moment Directed Edge Exploration Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Colin Cooper
    • 1
  • David Ilcinkas
    • 2
  • Ralf Klasing
    • 2
  • Adrian Kosowski
    • 2
    • 3
  1. 1.Dept of Computer ScienceKing’s College LondonUK
  2. 2.LaBRI, CNRS and Université de BordeauxFrance
  3. 3.Dept of Algorithms and System ModelingGdańsk University of TechnologyPoland

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