A Recursive Approach to Multi-robot Exploration of Trees

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 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 Ω ((log k)/ log log k). For k ≤ n robots the best algorithm known so far achieves a competitive time ratio of O (k/log k). Here, we improve this bound for trees with bounded depth or a minimum number of robots. Starting from a simple O(d)-competitive algorithm, called Yoyo, we recursively improve it by the Yo-star algorithm, which for any 0 < α < 1 transforms a g(d, k)-competitive algorithm into a O((g(d, k) log k + d1−α)(log k + logn))-competitive algorithm. So, we achieve a competitive bound of O ( 2O( √ (log d)(log log (log k)(log k + logn) ) . This improves the best known bounds for trees of depth d, whenever the number of robots is at least k = 2 √ (log d)(log log d)) and n = 2 √ log d).