Sensitivity and Block sensitivity

Question

May be this question is really silly and obvious but I am missing something subtle. I am reading on Sensitivity and Block sensitivity.

Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a Boolean function.

Let $[n]=\{1,2,\dots,n\}$.

If $i\in[n]$, let $\Bbb 1_i$ be length $n$ vector with all $0$s except $1$ at $i$th position.

If $B\subseteq [n]$, then $\Bbb 1_B$ be the length $n$ vector with $1$s only in positions marked by $B$.

If $i\in[n]$ and $x\in\{0,1\}^n$, let $x^i=x\oplus\Bbb 1_i$ where $\oplus$ is $XOR$ operation.

If $B\subseteq [n]$ and $x\in\{0,1\}^n$, let $x^{B}=x\oplus\Bbb 1_B$ where $\oplus$ is $XOR$ operation.

Sensitivity of $f$ at input $x$ is $$S_x(f) = |\{i:f(x)\neq f(x^i)\}|$$

Sensitivity of $f$ is $$S(f)=\max_xS_x(f)$$

Block Sensitivity of $f$ at input $x$, $BS_x(f)$ is maximum $r$ such that there is a set of disjoint subsets $\{B_i\}_{i=1}^r$($\forall i\neq j$, $B_i\cap B_j=\emptyset$) such that $$\forall j\mbox{, }f(x)\neq f(x^{B_j})$$

Block Sensitivity of $f$ is $$BS(f)=\max_xBS_x(f)$$

It is clear that $S(f)\leq BS(f)$ since we can take size $1$ blocks at coordinates of input $x$ that are modified to achieves $S(f)$.

So why can't I say $BS(f)\leq S(f)$? That is, suppose we found a maximum set of blocks $B_i$ that are disjoint, then cant I say:

1. There is no other bit position outside of $\cup_{i=1}^rB_i$ where we can change an input in order to change the input (Or else there is a size $1$ block that is disjoint and we can add to list $B_i$ incrasing $r$).

2. It is clear $S(f) = |\cup_{i=1}^rB_i|$ since if we change all bit positions in $B_i$ we change inputs and there is no other extra bit position by $1$.

3. Split the $B_i$ in to size $1$ blocks to get $S(f)=BS(f)$?

Am I wrong in the definition that $2$ DOES NOT HOLD since we can only consider $1$ block at a time and hence we can only say $S(f)\geq max_i|B_i|$? Anything else I am missing?

Is there any example for which $BS(f)\leq S(f)^{1+\epsilon}$?

Answer 1

2 does not hold as stated. All you know is that if you flip the all bits in a given block $B_i$, the output flips. Without additional conditions, I don't think you're guaranteed anything about flipping any individual bit within $B_i$.

EDIT: In fact, if $B_1, B_2, \ldots, B_i, \ldots$ represents a maximal set of blocks, then for any $B_i$, there cannot be more than one index within $B_i$ such that flipping the bit at that index will flip the output. Suppose that within $B_i$ there are two indices, say $j$ and $k$, such that flipping either $x_j$ or $x_k$ will flip the output. Then $B_i$ can be replaced by the two singletons $j$ and $k$, thus contradicting the assumption that the set of blocks was maximal.

Meanwhile, the first function found to have a quadratic gap between $BS(f)$ and $S(f)$ is due to Rubinstein (1995). You can find a quick summary of the construction in this blog post by Scott Aaronson. A function with slightly bigger separation was found later by Virza (2011).

Answer 2

Virza, improving on earlier work of Rubinstein, gives a function on $(2k+1)^2$ variables with sensitivity $2k+1$ and block sensitivity $(2k+1)(k+1)$. His function is defined as follows. Partition the input variables into $2k+1$ "sections" of size $2k+1$. The function is 1 if at least one section $x_1,\ldots,x_{2k+1}$ satisfies one of the following: (1) $x_{2i-1} = x_{2i} = 1$ for some $1 \leq i \leq k$ and all other $x_j$ are zero, or (2) $x_{2k+1} = 1$ and all other $x_j$ are zero.