Deterministic Meeting of Sniffing Agents in the Plane

Two mobile agents, starting at arbitrary, possibly different times from arbitrary locations in the plane, have to meet. Agents are modeled as discs of diameter 1, and meeting occurs when these discs touch. Agents have different labels which are integers from the set \(\{0,\dots ,L-1\}\). Each agent knows L and knows its own label, but not the label of the other agent. Agents are equipped with compasses and have synchronized clocks. They make a series of moves. Each move specifies the direction and the duration of moving. This includes a null move which consists in staying inert for some time, or forever. In a non-null move agents travel at the same constant speed, normalized to 1.

Agents have sensors enabling them to estimate the distance from the other agent, but not the direction towards it. We consider two models of estimation. In both models an agent reads its sensor at the moment of its appearance in the plane and then at the end of each move. This reading (together with the previous ones) determines the decision concerning the next move. In both models the reading of the sensor tells the agent if the other agent is already present. Moreover, in the monotone model, each agent can find out, for any two readings in moments \(t_1\) and \(t_2\), whether the distance from the other agent at time \(t_1\)was smaller, equal or larger than at time \(t_2\). In the weaker binary model, each agent can find out, at any reading, whether it is at distance less than \(\rho \) or at distance at least \(\rho \) from the other agent, for some real \(\rho >1\) unknown to them. Such distance estimation mechanism can be implemented, e.g., using chemical sensors. Each agent emits some chemical substance (scent), and the sensor of the other agent detects it, i.e., sniffs. The intensity of the scent decreases with the distance. In the monotone model it is assumed that the sensor is ideally accurate and can measure any change of intensity. In the binary model it is only assumed that the sensor can detect the scent below some distance (without being able to measure intensity) above which the scent is too weak to be detected.

We show the impact of the two ways of sensing on the time of meeting, measured from the start of the later agent. For the monotone model we show an algorithm achieving meeting in time O(D), where D is the initial distance between the agents. This complexity is optimal. For the binary model we show that, if agents start at distance smaller than \(\rho \) (i.e., when they sense each other initially) then meeting can be guaranteed within time \(O(\rho \log L)\), and that this time cannot be improved in general. Finally we observe that, if agents start at distance \(\alpha \rho \), for some constant \(\alpha >1\) in the binary model, then sniffing does not help, i.e., the worst-case optimal meeting time is of the same order of magnitude as without any sniffing ability.