Finding paths in sparse random graphs requires many queries

Abstract

AbstractWe 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 in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in when for some fixed constant . This random graph is known to have typically linearly long paths. To have edges with high probability in one clearly needs to query at least 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 with at least constant probability in with must query at least pairs of vertices. This is tight up to the factor. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 71–85, 2017