A Recursive Approach to Multi-robot Exploration of Trees

Abstract

The multi-robot exploration problem is to explore an unknown graph of size n and depth d with k robots starting from the same node. For known graphs a traversal of all nodes takes at most ${\cal O}(d + n/k)$ steps. The ratio between the time until cooperating robots explore an unknown graph and the optimal traversal of a known graph is called the competitive exploration time ratio.

It is known that for any algorithm this ratio is at least $\Omega\left((\log k)/\log \log k\right)$. For k ≤ n robots the best algorithm known so far achieves a competitive time ratio of ${\cal O}\left({k}/{\log k} \right)$.

Here, we improve this bound for trees with bounded depth or a minimum number of robots. Starting from a simple ${\cal O}(d)$-competitive algorithm, called Yo-yo, we recursively improve it by the Yo-star algorithm, which for any 0 < α < 1 transforms a g(d,k)-competitive algorithm into a ${\cal O}( (g(d^{\alpha},k) \log k + d^{1-\alpha})(\log k + \log n))$-competitive algorithm. So, we achieve a competitive bound of ${\cal O}\left(2^{{\cal O}(\sqrt{(\log d)(\log\log k)})}(\log k)(\log k+ \log n)\right)$. This improves the best known bounds for trees of depth d, whenever the number of robots is at least $k=2^{\omega(\sqrt{(\log d)(\log \log d)})}$ and $n=2^{O(2^{\sqrt{\log d}})}$.