Complexité en espace de l'exploration de graphes

Graph Exploration is a problem motivated by several aspects of theoretical computer science, including logic and computational complexity. It has also numerous applications in robotics. In all these contexts, the amount of memory used by the mobile entity (robot, finite automaton, etc.) performing exploration is a key parameter. In this thesis, we study in depth the space complexity of graph exploration, in two different frameworks.In the first part of the thesis, we focus on exploration ”without assistance”, i.e., the case where the mobile entity does not possess any a priori information about the explored graph. In this context, we prove several lower and upper bounds on the amount of memory necessary and sufficient to the mobile entity for performing exploration of all graphs. In particular, we show that the depth-first search algorithm is optimal in space when the complexity is measured in term of degree and diameter. In the second part of the thesis, we focus on exploration ”with assistance”, by assuming that an oracle entirely aware of the input graph can help the mobile entity, by providing some information about it. This information is measured by its bit-space complexity, and we are interested in the minimal number of bits that the oracle must provide so that the mobile entity can perform exploration. The information can be either given directly to the entity, or encodedon the vertices of the graph.