# Optimal Separation and Strong Direct Sum for Randomized Query Complexity

## TL;DR

This paper proves new theorems on the query complexity of randomized algorithms, establishing bounds and direct sum properties, and demonstrates their implications for communication complexity, answering longstanding open questions.

## Contribution

It introduces a bounded-error separation theorem and a strong direct sum theorem for randomized query complexity, advancing understanding of complexity bounds and their applications.

## Key findings

- Existence of a total function with ( ext{R}(f) \, ext{log}(1/\u03b5)) query complexity
- Optimal superlinear direct-sum theorem for randomized query complexity
- Implications for communication complexity, confirming open questions

## Abstract

We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : \{0,1\}^n \to \{0,1\}$ whose $\epsilon$-error randomized query complexity satisfies $\overline{\mathrm{R}}_\epsilon(f) = \Omega( \mathrm{R}(f) \cdot \log\frac1\epsilon)$. * Strong direct sum theorem. For every function $f$ and every $k \ge 2$, the randomized query complexity of computing $k$ instances of $f$ simultaneously satisfies $\overline{\mathrm{R}}_\epsilon(f^k) = \Theta(k \cdot \overline{\mathrm{R}}_{\frac\epsilon k}(f))$. As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function $f$ for which $\mathrm{R}(f^k) = \Theta( k \log k \cdot \mathrm{R}(f))$. This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of G\"o\"os, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies $\mathrm{R}^{\mathrm{cc}} (f^k) = \Theta( k \log k \cdot \mathrm{R}^{\mathrm{cc}}(f))$, answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).