Haystack Hunting Hints and Locker Room Communication

TL;DR

This paper investigates how minimal advice can significantly improve the probability of finding a specific object in a large unstructured set modeled by a permutation, establishing bounds on success probabilities.

Contribution

It introduces a model where one bit of advice improves search success in permutations and derives tight bounds on success probabilities.

Findings

One bit of advice increases success probability by a logarithmic factor.

Established asymptotically matching upper and lower bounds for success probabilities.

Linked permutation properties to the problem, leveraging properties related to the rencontres number.

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.