Almost Optimal Explicit Selectors

摘要

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 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 O(min [n, ((k~2)/(k-r+1) polylog n]), for any parameters r ≤ k ≤ n. We establish a lower bound Ω(min [n, (k~2)/(k-r+1) • (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.