Constructive derandomization of query algorithms

TL;DR

This paper presents efficient methods to convert randomized query algorithms into deterministic ones with near-optimal parameters, enabling derandomization and dequantization of algorithms in decision problems.

Contribution

The paper introduces near-optimal deterministic algorithms for derandomizing query algorithms, including instance-optimal and online variants, and applies these techniques to classical complexity results.

Findings

Deterministic algorithms with near-optimal query complexity and runtime.

Instance-optimal and online derandomization algorithms.

Constructive relations between classical complexity measures.

Abstract

We give efficient deterministic algorithms for converting randomized query algorithms into deterministic ones. We first give an algorithm that takes as input a randomized $q$-query algorithm $R$ with description length $N$ and a parameter $\varepsilon$, runs in time $\mathrm{poly}(N) \cdot 2^{O(q/\varepsilon)}$, and returns a deterministic $O(q/\varepsilon)$-query algorithm $D$ that $\varepsilon$-approximates the acceptance probabilities of $R$. These parameters are near-optimal: runtime $N + 2^{\Omega(q/\varepsilon)}$ and query complexity $\Omega(q/\varepsilon)$ are necessary. Next, we give algorithms for instance-optimal and online versions of the problem: $\circ$ Instance optimal: Construct a deterministic $q^\star_R$-query algorithm $D$, where $q^\star_R$ is minimum query complexity of any deterministic algorithm that $\varepsilon$-approximates $R$. $\circ$ Online: Deterministically approximate the acceptance probability of $R$ for a specific input $\underline{x}$ in time $\mathrm{poly}(N,q,1/\varepsilon)$, without constructing $D$ in its entirety. Applying the techniques we develop for these extensions, we constructivize classic results that relate the deterministic, randomized, and quantum query complexities of boolean functions (Nisan, STOC 1989; Beals et al., FOCS 1998). This has direct implications for the Turing machine model of computation: sublinear-time algorithms for total decision problems can be efficiently derandomized and dequantized with a subexponential-time preprocessing step.