Efficient Exploration of Faulty Trees

We consider the problem of the exploration of trees, some of whose edges are faulty. A robot, situated in a starting node and unaware of the location of faults, has to explore the connected fault-free component of this node by visiting all its nodes. The cost of the exploration is the number of edge traversals. For a given tree and given starting node, the overhead of an exploration algorithm is the worst-case ratio (taken over all fault configurations) of its cost to the cost of an optimal algorithm which knows where faults are situated. An algorithm, for a given tree and given starting node, is called perfectly competitive if its overhead is the smallest among all exploration algorithms not knowing the location of faults. We design a perfectly competitive exploration algorithm for any line, and an exploration algorithm for any tree, whose overhead is at most 9/8 larger than that of a perfectly competitive algorithm. Both our algorithms are fairly natural and the total time of local computations used during exploration is linear in the size of the explored tree. Our main contribution is the analysis of the performance of these algorithms, showing that natural exploration strategies perform well in faulty trees.