Almost Optimal Explicit Selectors

Abstract

We understand selection by intersection as distinguishing a single element of a set by the uniqueness of its occurrence in some other set. More precisely, given two sets A and B, if A ∩ B = {z}, then element z∈ A is selected by set B. Selectors are such families $\mathcal{S}$ of sets B of some domain that allow to select many elements from sufficiently small subsets A of the domain. Selectors are used in communication protocols for the multiple-access channel, in implementations of distributed-computing primitives in radio networks, and in algorithms for group testing. We give new explicit (n,k,r)-selectors of size $\mathcal{O}(min [n, \frac{k^2}{k-r+1} polylog~n])$, for any parameters r ≤ k ≤ n. We establish a lower bound $\Omega(min [n, \frac{k^2}{k-r+1} \cdot \frac{log(n/k)}{log(k/(k-r+1))}])$ on the length of (n,k,r)-selectors, which demonstrates that our construction is within a polylog n factor close to optimal. The new selectors are applied to develop explicit implementations of selection resolution on the multiple-access channel, gossiping in radio networks and an algorithm for group testing with inhibitors.