Derandomizing random walks in undirected graphs using locally fair exploration strategies

Colin Cooper (1) , David Ilcinkas (2, 3) , Ralf Klasing (2, 3) , Adrian Kosowski (2, 3)

1 DCS - Department of Computer Science
2 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms
3 Combinatoire et Algorithmique

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 (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 diam within time O(diam|E|), and in the long run traverses all edges of G with the same frequency.

Mots clés: Random walk; Graph exploration

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Algorithme et structure de données [cs.DS]

Colin Cooper, David Ilcinkas, Ralf Klasing, Adrian Kosowski. Derandomizing random walks in undirected graphs using locally fair exploration strategies. Distributed Computing, 2011, 24 (2), pp.91-99. ⟨10.1007/s00446-011-0138-4⟩. ⟨hal-00638229⟩