  Consider a family of sets and a single set, called the query set. How can one
quickly find a member of the family which has a maximal intersection with the
query set? Time constraints on the query and on a possible preprocessing of the
set family make this problem challenging. Such maximal intersection queries
arise in a wide range of applications, including web search, recommendation
systems, and distributing on-line advertisements. In general, maximal
intersection queries are computationally expensive. We investigate two
well-motivated distributions over all families of sets and propose an algorithm
for each of them. We show that with very high probability an almost optimal
solution is found in time which is logarithmic in the size of the family.
Moreover, we point out a threshold phenomenon on the probabilities of
intersecting sets in each of our two input models which leads to the efficient
algorithms mentioned above.
