Collective Tree Exploration via Potential Function Method

We study the problem of collective tree exploration (CTE) in which a team of k agents is tasked to traverse all the edges of an unknown tree as fast as possible, assuming complete communication between the agents [FGKP06]. In this paper, we present an algorithm performing collective tree exploration in 2n/k+𝒪(kD) rounds, where n is the number of nodes in the tree, and D is the tree depth. This leads to a competitive ratio of 𝒪(√k), the first polynomial improvement over the 𝒪(k) ratio of depth-first search. Our analysis holds for an asynchronous generalization of collective tree exploration. It relies on a game with robots at the leaves of a continuously growing tree extending the "tree-mining game" of [C23] and resembling the "evolving tree game" of [BCR22]. Another surprising consequence of our results is the existence of algorithms {𝒜_k}_{k ∈ ℕ} for layered tree traversal (LTT) with cost at most 2L/k+𝒪(kD), where L is the sum of all edge lengths. For the case of layered trees of width w and unit edge lengths, our guarantee is thus in 𝒪(√wD).