International audienceA finite automaton, simply referred to as a {\em robot}, has to explore a graph, i.e., visit all the nodes of the graph. The robot has no a priori knowledge of the topology of the graph or of its size. It is known that, for any $k$-state robot, there exists a $(k+1)$-node graph of maximum degree~3 that the robot cannot explore. This paper considers the effects of allowing the system designer to add short labels to the graph nodes in a preprocessing stage, and using these labels to guide the exploration by the robot. We describe an exploration algorithm that given appropriate 2-bit labels (in fact, only 3-valued labels) allows a robot to explore all graphs. Furthermore, we describe a suitable labeling algorithm for generating the required labels, in linear time. We also show how to modify our labeling scheme so that a robot can explore all graphs of bounded degree, given appropriate 1-bit labels. In other words, although there is no robot able to explore all graphs of maximum degree~3, there is a robot $\cR$, and a way to color in black or white the nodes of any bounded-degree graph $G$, so that $\cR$ can explore the colored graph $G$. Finally, we give impossibility results regarding graph exploration by a robot with no internal memory ({\it i.e.}, a single state automaton)

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ > 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O((d r) δ polylog(d r)), where r = min(r1, r2) and for r ≥ 1. 1 Introduction 1.1 The problem and the model Consider a Euclidean δ-dimensional space F. We construct an infinite grid G δ of dimension δ as follows. The set of nodes of G δ are the points of F with ⋆ Partially supported by the ANR project ALADDIN, the INRIA project CEPAGE and by a France-Israel cooperation grant (Multi-Computing project).