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 G ∼ G(n, p) in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in G ∼ G(n, p) when p = 1+ε n for some fixed constant ε > 0. This random graph is known to have typically linearly long paths. To have ` edges with high probability in G ∼ G(n, p) one clearly needs to query at least Ω ( ` p ) pairs of vertices. Can we find a path of length ` 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 ` = Ω ( log( 1ε ) ε ) with at least constant probability in G ∼ G(n, p) with p = 1+ε n must query at least Ω ( ` pε log( 1ε ) ) pairs of vertices. This is tight up to the log ( 1 ε ) factor.