Fast collaborative graph exploration

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 $k = D n^{1+ \epsilon} < n^{2+\epsilon}$, for any $\epsilon > 0$. 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 $c>1$, we show that in the global communication model, a team of $k = D n^c$ 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.