Uniform Universal Sets, Splitters, and Bisectors

TL;DR

This paper introduces new derandomized structures called bisectors and universal sets that optimize probabilistic algorithms and complexity class reductions, using simple, efficient constructions.

Contribution

It constructs bisectors and universal sets with near-optimal size and linear time complexity, simplifying previous complex combinatorial methods.

Findings

Constructed bisectors of size 2^{k+o(k)} that derandomize probabilistic search.

Developed universal sets that are also bisectors, with optimal size and linear construction time.

Enabled derandomization of reductions between average case complexity classes.

Abstract

Given a subset of size $k$ of a very large universe a randomized way to find this subset could consist of deleting half of the universe and then searching the remaining part. With a probability of $2^{-k}$ one will succeed. By probability amplification, a randomized algorithm needs about $2^k$ rounds until it succeeds. We construct bisectors that derandomize this process and have size~$2^{k+o(k)}$. One application is derandomization of reductions between average case complexity classes. We also construct uniform $(n,k)$-universal sets that generalize universal sets in such a way that they are bisectors at the same time. This construction needs only linear time and produces families of asymptotically optimal size without using advanced combinatorial constructions as subroutines, which previous families did, but are basedmainly on modulo functions and refined brute force search.