Brief Announcement: Efficient Collaborative Tree Exploration with Breadth-First Depth-Next

We consider the problem of collaborative tree exploration posed by Fraigniaud, Gasieniec, Kowalski, and Pelc [8] where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible and return to the root. Denoting by.. the total number of nodes and by.. the tree depth, the O(n/log(k) +D) algorithm of [8] achieves the best competitive ratio known with respect to the optimal exploration algorithm that knows the tree in advance, which takes order max {2n/k, 2D} rounds. Brass, Cabrera-Mora, Gasparri, and Xiao [1] consider an alternative performance criterion, the additive overhead with respect to 2n/k, and obtain a 2n/k + O( (D +k)(k)) runtime guarantee. In this announcement, we present 'Breadth-First Depth-Next' (BFDN), a novel and simple algorithm that performs collaborative tree exploration in time 2n/k + O(D-2 log(k)), thus outperforming [1] for all values of (n,D) and being order-optimal for fixed.. and trees with depth D = o (root n). The proof of our result crucially relies on the analysis of a simple two-player game with balls in urns that could be of independent interest. 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)D1+1/ l) for parameter l >= 1, thereby improving performance for trees with large depth. A complete version of the paper is available online [2].