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Uniform Universal Sets, Splitters, and Bisectors

Elisabet Burjons, Peter Rossmanith

TL;DR

The paper tackles derandomization by constructing three interconnected families—uniform splitters, uniform bisectors, and uniform universal sets—that mimic probabilistic behavior in a deterministic, linear-time manner. It introduces and leverages modulo-based constructions and the Chinese Remainder Theorem to achieve near-optimal sizes: uniform splitters with size $O(k^4 ext{log }n/ ext{log } ext{ t elax ell})$ for $ ext{ t elax ell}\ge k^3$, and uniform bisectors of size $2^{k+o(k)}$ independent of $n$, along with uniform $(n,k, ext{ t elax alpha})$-universal sets of size $(1/ ext{ t elax alpha})^{k}k^{O(k^{5/6})} ext{log }n$. The constructions avoid heavy combinatorial subroutines and yield linear-time procedures, enabling applications in derandomization of reductions and average-case complexity analyses. The work also provides a framework of mapping families to extend universal- and bisector-like properties and discusses open questions about tightening bounds and balancing trade-offs. Overall, the results advance deterministic derandomization by delivering simple, scalable, and near-optimal uniform combinatorial tools with broad algorithmic implications.

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.

Uniform Universal Sets, Splitters, and Bisectors

TL;DR

The paper tackles derandomization by constructing three interconnected families—uniform splitters, uniform bisectors, and uniform universal sets—that mimic probabilistic behavior in a deterministic, linear-time manner. It introduces and leverages modulo-based constructions and the Chinese Remainder Theorem to achieve near-optimal sizes: uniform splitters with size for , and uniform bisectors of size independent of , along with uniform -universal sets of size . The constructions avoid heavy combinatorial subroutines and yield linear-time procedures, enabling applications in derandomization of reductions and average-case complexity analyses. The work also provides a framework of mapping families to extend universal- and bisector-like properties and discusses open questions about tightening bounds and balancing trade-offs. Overall, the results advance deterministic derandomization by delivering simple, scalable, and near-optimal uniform combinatorial tools with broad algorithmic implications.

Abstract

Given a subset of size 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 one will succeed. By probability amplification, a randomized algorithm needs about rounds until it succeeds. We construct bisectors that derandomize this process and have size~. One application is derandomization of reductions between average case complexity classes. We also construct uniform -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.
Paper Structure (6 sections, 18 theorems, 4 equations, 1 figure)

This paper contains 6 sections, 18 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Uniform $(n,k,\ell)$-splitters
  4. Bisectors
  5. Uniform Universal Sets
  6. Conclusion

Key Result

Lemma 5

Let $\cal F$ be an $a$-uniform $(n,k,\ell)$-splitter. If $n\geq a\ell(k+1)$ then we can construct from $\cal F$ a uniform $(n,k,\ell)$-splitter $\cal F'$. The time to compute $\cal F'$ from $\cal F$ is linear and $|{\cal F'}|=(k+1)|\cal F|$.

Figures (1)

  • Figure 2: Two strategies to construct a bisector. In the left picture we start with a function that maps all elements to $0$ and on each step we map some elements to $1$ leaving out a potential $k$-set. In the right picture, we start with a function that maps all elements to $1$ and in every step we make sure that a significat part of a given a set $S$ is mapped to $0$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5: Smoothing Lemma
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • Definition 11
  • ...and 15 more