lec17.md

Asymptotics I

• Pick some representative operation → cost model
• Big O: Bounding above (less than)
• Big 𝛀: Bounding below (greater than)
• Big 𝚯: Bounding above & below (equals)

Big 𝚯

• Performance measurement $$R(N)$$, $$N = \text{size of problem}$$

$$ R(N) \in \Theta(f(N)) \implies \exists\ \ k_1,\ k_2 > 0 \ |\ k_1 \cdot f(N) \leq R(N) \leq k_2 \cdot f(N) $$ $$ \forall\ N > N_0 $$

Big O

• Only bounds from above
• Order of growth of $$R(N) \leq f(N)$$

$$ R(N) \in O(f(N)) \implies \exists\ \ k > 0 \ |\ R(N) \leq k \cdot f(N) $$ $$ \forall\ N > N_0 $$

Summary

Category	Informal Meaning	Family	Family Members
Big 𝚯 $$\Theta(f(N))$$	$$R(N) \propto f(N)$$	$$\Theta(N^2)$$	$$\frac{N^2}{2}$$ $$2N^2$$ $$N^2 + 38N + N$$
Big O $$O(f(N))$$	$$R(N) \leq f(N)$$	$$O(N^2)$$	$$\frac{N^2}{2}$$ $$2N^2$$ $$\log(N)$$
Big 𝛀 $$\Omega(f(N))$$	$$R(N) \geq f(N)$$	$$\Omega(N^2)$$	$$\frac{N^2}{2}$$ $$2N^2$$ $$e^N$$
Tilde $$\sim f(N)$$	$$\lim_{n \to \infty} \frac{R(N)}{f(N)} = 1$$	$$\sim 2N^2$$	$$2N^2$$ $$2N^2 + 5$$