Want to Gather? No Need to Chatter!

Abstract

A team of mobile agents with different labels, starting from different nodes of an unknown anonymous network, must meet at the same node and declare that they all met. This task of gathering was traditionally considered assuming that agents at the same node can exchange information. We ask if this ability of talking is needed. The answer turns out to be no. We design two deterministic algorithms that accomplish gathering in a much weaker model. We only assume that each agent knows how many agents are at the node that it currently occupies. Our first algorithm assumes that agents know some upper bound N on the size of the network, and works in time polynomial in N and in the length of the smallest label. Our second algorithm does not assume any knowledge about the network, but its complexity is at least exponential in the size of the network and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product we solve, in the same weak model, the fundamental problem of leader election among agents. As an application we solve the gossiping problem in this model: if each agent has a message, all agents can learn all messages. This is perhaps our most surprising finding: agents without any transmitting devices can solve the most general information exchange problem if they can count the number of agents present at visited nodes.