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Volume 14 (2018) Article 5 pp. 1-27
Randomized Query Complexity of Sabotaged and Composed Functions
Received: June 9, 2016
Revised: March 2, 2017
Published: May 4, 2018
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Keywords: randomized algorithms, query complexity, composition theorems
ACM Classification: F.1.2, F.1.3
AMS Classification: 68W20, 68Q15, 68Q17

Abstract: [Plain Text Version]

$ \newcommand{\R}{{\mathbb R}} \newcommand{\RS}{{\mathsf{RS}}} $

We study the composition question for bounded-error randomized query complexity: Is $\R(f \circ g) = \Omega(\R(f)\R(g))$ for all Boolean functions $f$ and $g$? We show that inserting a simple Boolean function $h$, whose query complexity is only $\Theta(\log \R(g))$, in between $f$ and $g$ allows us to prove $\R(f\circ h\circ g) = \Omega(\R(f)\R(h)\R(g))$.

We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, $\RS(f)$. Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, $\RS(f \circ g) \geq \RS(f) \RS(g)$, and a composition theorem with randomized query complexity, $\R(f \circ g) = \Omega(\R(f)\RS(g))$. It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity.

Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem for zero-error randomized protocols implies a general lifting theorem for bounded-error protocols.

A conference version of this paper appeared in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016).