Distributed Exploration of an Unknown Graph

Abstract

We consider a group of identical asynchronous agents, initially located at different nodes of an undirected simple graph. The nodes of the graph are unlabeled, and each contains a whiteboard where the visiting agents can write to and read from. The agents, initially, do not know the graph nor its topology. The only a-priori knowledge the agents may have is either the number n of nodes, or the total number k of agents. The goal is for the agents to construct a labelled map of the unknown graph, the same for all agents, so to be in complete agreement with each-other about their environment. This problem, called Labelled Map Construction, is closely related to a variety of other basic problems, including election and rendezvous. We are interested in efficient and generic protocols that can solve the problem, irrespective of the graph topology, where the cost of the algorithm is measured in terms of the total number of moves (or, edge traversals) made by the agents. We present a novel deterministic algorithm that, provided that n and k are co-prime (a necessary condition), constructs a map of the graph, elects a leader among the agents, and provides a unique labelling on the nodes of the graph. Our algorithm uses no more than O(km) edge traversals where m is the number of edges in the graph. Our result improves on the finding by Barriere et al. [4] for graphs with sense of direction, extending it to graphs with arbitrary labelling, provided that one of n or k is known.