Finding paths in sparse random graphs requires many queries

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph (Formula presented.) in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in (Formula presented.) when (Formula presented.) for some fixed constant (Formula presented.). This random graph is known to have typically linearly long paths. To have (Formula presented.) edges with high probability in (Formula presented.) one clearly needs to query at least (Formula presented.) pairs of vertices. Can we find a path of length (Formula presented.) economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length (Formula presented.) with at least constant probability in (Formula presented.) with (Formula presented.) must query at least (Formula presented.) pairs of vertices. This is tight up to the (Formula presented.) factor.