Haystack hunting hints and locker room communication

We want to efficiently find a specific object in a large unstructured set, which we model by a random n$$ n $$‐permutation, and we have to do it by revealing just a single element. Clearly, without any help this task is hopeless and the best one can do is to select the element at random, and achieve the success probability 1n$$ \frac{1}{n} $$ . Can we do better with some small amount of advice about the permutation, even without knowing the target object? We show that by providing advice of just one integer in {0,1,…,n−1}$$ \left\{0,1,\dots, n-1\right\} $$ , one can improve the success probability considerably, by a Θ(lognloglogn)$$ \Theta \left(\frac{\log n}{\mathrm{loglog}n}\right) $$ factor. We study this and related problems, and show asymptotically matching upper and lower bounds for their optimal probability of success. Our analysis relies on a close relationship of such problems to some intrinsic properties of random permutations related to the rencontres number.