Breadth-First Depth-Next: Optimal Collaborative Exploration of Trees with Low Diameter
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We consider the problem of collaborative tree exploration posed by Fraigniaud, Gasieniec, Kowalski, and Pelc where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible. Denoting by n the total number of nodes and by D the tree depth, the 𝒪(n/log(k)+D) algorithm of Fraigniaud et al. achieves the best-known competitive ratio with respect to the cost of offline exploration which is Θ(max{2n/k,2D}). Brass, Cabrera-Mora, Gasparri, and Xiao consider an alternative performance criterion, namely the additive overhead with respect to 2n/k, and obtain a 2n/k+𝒪((D+k)^k) runtime guarantee. In this paper, we introduce `Breadth-First Depth-Next' (BFDN), a novel and simple algorithm that performs collaborative tree exploration in time 2n/k+𝒪(D^2log(k)), thus outperforming Brass et al. for all values of (n,D) and being order-optimal for all trees with depth D=o_k(√(n)). Moreover, a recent result from Disser et al. implies that no exploration algorithm can achieve a 2n/k+𝒪(D^2-ϵ) runtime guarantee. The dependency in D^2 of our bound is in this sense optimal. The proof of our result crucially relies on the analysis of an associated two-player game. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of 𝒪_ℓ(n/k^1/ℓ+log(k) D^1+1/ℓ) for parameter ℓ≥ 1, thereby improving performance for trees with large depth.
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