Assigning labels in an unknown anonymous network with a leader

We consider the task of assigning distinct labels to nodes of an unknown anonymous network in a distributed manner. A priori, nodes do not have any identities, except for one distinguished node, called the source, and do not know the topology or the size of the network. They execute identical algorithms, apart from the source which plays the role of a leader and starts the labeling process. Our goal is to assign short labels, as fast as possible. The quality of a labeling algorithm is measured by the range from which the algorithm picks the labels, or alternatively, the length of the assigned labels. Natural efficiency measures are the time, i.e., the number of rounds required for the label assignment, and the message and bit complexities of the label assignment protocol, i.e., the total number of messages (resp., bits) circulating in the network. We present label assignment algorithms whose time and message complexity are asymptotically optimal and which assign short labels. On the other hand, we establish inherent tradeoffs between quality and efficiency for labeling algorithms.