1 concrete complexity

1.1 decision trees

• internal nodes labelled with input vars
• leaves are labelled with output value

They compute functions f: \{0, 1\}^n \to \{0, 1\}

D(f) = \text{least height of a DT computing } f

1.1.1 certificates

Partial assignment \rho \in \{0, 1, *\} is a certificate for x \in \{0, 1\}^n if

• \rho is consistent with x: \forall i \rho_i \ne * \implies \rho_i = x_i
• \rho is consistent with x' \implies f(x) = f(x')

C(f, x) = \text{least size of certificate for } x

then

C_b(f) = \max_{x: f(x) = b} C(f, x)

C(f) = \max_x C(f, x) = \max\{C_0(f), C_1(f)\}

C(f) \le D(f)

C_1(f) = \text{least } k \text{ such that } f \text{ can be written as } k\text{-DNF}

C_0(f) = \text{least } k \text{ such that } f \text{ can be written as } k\text{-CNF}

D(f) \le C_1(f)C_0(f) \le C^2(f)

1.1.2 sensitivity

Let x^{(i)} be x but with x_i flipped. Then

s(f, x) = |\{i : f(x) \ne f(x^{(i)})\}|

s(f) = \max_x s(f, x)

s(f) \le C(f)

D(f) \le s^k(f) \text{ for some } k

1.1.2.1 block sensitivity

bs(f, x) = max k of disjoint B_1, \cdots, B_k \subseteq [n] such that f(x) \ne f(x^{B_i})

C(f) \le s(f) \cdot bs(f)