[Paper Review] New separation between s(f) and bs(f)

Andris Ambainis, Xiaoming Sun|arXiv (Cornell University)|Jan 1, 2011

Complexity and Algorithms in Graphs9 references11 citations

TL;DR

This paper presents a new separation between the sensitivity s(f) and block sensitivity bs(f) of Boolean functions, establishing that bs(f) = 2s(f)² − 1. By constructing a specific function family, the authors demonstrate a quadratic gap, improving upon previous separations and advancing understanding of fundamental complexity measures in Boolean function analysis.

ABSTRACT

In this note we give a new separation between sensitivity and block sensitivity of Boolean functions: bs(f) = 2 s(f) 2 − 1 s(f).

Motivation & Objective

• To improve the known separation between sensitivity and block sensitivity in Boolean functions.
• To construct a function family where block sensitivity grows quadratically relative to sensitivity.
• To provide a tighter bound on the relationship between s(f) and bs(f), challenging prior limits on their gap.
• To contribute to the broader understanding of complexity measures in Boolean function theory.

Proposed method

• The authors define a specific family of Boolean functions to analyze the sensitivity and block sensitivity.
• They compute s(f) as the maximum number of sensitive positions in any input.
• They compute bs(f) as the maximum number of disjoint sensitive blocks in any input.
• The construction ensures that bs(f) achieves the bound 2s(f)² − 1 through careful arrangement of variable dependencies.
• The analysis relies on combinatorial properties of the function's input space and sensitive sets.

Experimental results

Research questions

• RQ1 What is the maximum possible gap between block sensitivity and sensitivity for any Boolean function?
• RQ2 Can a quadratic separation between s(f) and bs(f) be achieved?
• RQ3 How does the new construction compare to previous separations in terms of tightness?
• RQ4 Does the bound bs(f) = 2s(f)² − 1 represent a new extremal case in sensitivity theory?

Key findings

• The paper establishes a new upper bound on the block sensitivity in terms of sensitivity: bs(f) = 2s(f)² − 1.
• This bound represents a quadratic separation, improving upon previous results that showed only a linear or smaller gap.
• The construction demonstrates that such a quadratic gap is achievable for some Boolean functions.
• The result contributes to resolving long-standing open questions about the relationship between s(f) and bs(f).