Fast Collaborative Graph Exploration

Abstract

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 = Dn 1+ < n 2+ , for any > 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 = Dn c agents can always complete exploration in D(1 + 1 c−1 + o(1)) time steps, whereas at least D(1 + 1 c − o(1)) steps are sometimes required. In the local communication model, D(1 + 2 c−1 + o(1)) steps always suffice to complete exploration, and at least D(1 + 2 c − o(1)) steps are sometimes required. This shows a clear separation between the global and local communication models.

A general lower bound for collaborative tree exploration

“…Secondly, we show that for every constant ε > 0, there is a constant D = D(ε) such that for any exploration algorithm with k ≤ Dn 1+ε agents, there exists a tree in T n,D on which the algorithm needs at least D/(5ε) rounds. This (almost) tightly matches the algorithm of Dereniowski et al [12], which can explore any tree in at most (1 + o(1))D/ε rounds using k = Dn 1+ε agents. Our result implies that any exploration algorithm with k = Dn 1+o(1) agents has competitive ratio ω(1).…”

“…The lower bounds for collaborative tree exploration discussed above carry over to the collaborative exploration of general undirected graphs with distinguishable vertices. Also, the algorithm of Dereniowski et al [12] for k = Dn 1+ε works on general graphs. Additionally, Ortolf and Schindelhauer [22] gave a lower bound on the best-possible competitive ratio for randomized algorithms of Ω( √ log k/ log log k) for k = √ n. Collaborative exploration by multiple random walks without communication has been considered by Alon et al [3], Elsässer and Sauerwald [17], and Ortolf and Schindelhauer [24].…”

“…Surprisingly, Dereniowski et al [12] showed that already a polynomial number k = Dn 1+ε of agents allows for a BFS-like algorithm that achieves a constant competitive ratio. For smaller teams of agents, Fraigniaud et al [18,20] gave a collaborative algorithm with competitive ratio O(k/ log k).…”

Beachcombing on Strips and Islands

“…Many searching and exploration algorithms are studied in the online setting, i.e., the target position or sometimes other parameters of the environment are a priori unknown (cf. [2,3,9,14,16,19,20]). Efficiency of such algorithms is typically measured by the competitive ratio, i.e., the ratio of the time spent by the online algorithm with respect to the time of the optimal offline algorithm.…”

“…Sets of collaborating mobile robots were studied, e.g., in [10,15,22,23]. Tradeoffs between the number of robots and the time of exploration were derived in [19].…”