Fast collaborative graph exploration

• Authors: Dariusz Dereniowski;Yann Disser;Adrian Kosowski;Dominik Pająk;Przemysław Uznański
• Venue: ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
• Year: 2013

We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k=Dn1+εn2+ε, for any ε0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D. We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large k. For any constant c1, we show that in the global communication model, a team of k=Dnc agents can always complete exploration in $D(1+ \frac{1}{c-1} +o(1))$ time steps, whereas at least $D(1+ \frac{1}{c} -o(1))$ steps are sometimes required. In the local communication model, $D(1+ \frac{2}{c-1} +o(1))$ steps always suffice to complete exploration, and at least $D(1+ \frac{2}{c} -o(1))$ steps are sometimes required. This shows a clear separation between the global and local communication models.