Haystack Hunting Hints and Locker Room Communication

Abstract

We want to efficiently find a specific object in a large unstructured set, which we model by a random $n$-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 select the element at random, and achieve the success probability $\frac{1}{n}$. Can we do better with some small amount of advice about the permutation, even without knowing the object sought? We show that by providing advice of just one integer in $\{0,1,...,n-1\}$, one can improve the success probability considerably, by a $\Theta (\frac{logn}{loglogn})$ 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 rendom permutations related to the rencontres number.

27 pages, 1 figure