Big-Omega & Big-Theta

Big-Oh Gives Upper Bounds

For f,g functions $\N \rightarrow \R^+$,
$f(n) = \Omega(g(n))$ means there are $c,n_0 > 0$ s.t. $\forall n>n_0\quad f(n) \geq c \times g(n)$

i.e. f is asymptotically bounded from below by g

(A graph with two lines. f is a blue line with a wobbly, but mostly linear movement upwards. c times g is a red line which has a similar trajectory, but end up just slightly below the blue line.)

Note: We may have $C << 1$

or f grows asymptotically at least as fast as g.

Big-Oh & Big-Omega are Duals

Fact: $f(n) = \Omega(g(n)) \leftrightarrow g(n) = O(f(n))$

Pf: $f(n) = \Omega(g(n))$:

• $$\iff \exists n_0, c' > 0 \text{ s.t. } n>n_0 \implies f(n) \geq c' \cdot g(n)$$
• $$\iff \exists n_0,c' > 0 \text{ s.t. } n>n_0 \implies c'\cdot g(n) \leq f(n)$$
• $\iff\exists n_0,c > 0 \text{ s.t. } n>n_0 \implies g(n) \leq c\cdot f(n)$ // letting $c = \frac{1}{c'}$
• $$\iff g(n) O(f(n))$$

So: f grows at least as fast as g $\iff$ g grows at most as fast as f.

Examples: Worse-case times

Operation	$O(1)$	$\Omega(1)$	$O(\log n)$ (highlighted)	$\Omega(\log n)$ (highlighted)	$O(n)$	$\Omega(n)$	$O(n \log n)$	$\Omega(n \log n)$
stack push/pop (highlighted)	✓ (green)	✓ (green)	✓	❌ (blue)	✓	❌ (blue)	✓	❌ (blue)
unqueue/dequeue	✓ (green)	✓ (green)	✓	❌ (blue)	✓	❌ (blue)	✓	❌ (blue)
heap insert or extract min	❌ (pink)	✓	✓ (green)	✓ (green)	✓	❌ (blue)	✓	❌ (blue)
AVL-tree find, insert, remove	❌ (pink)	✓	✓ (green)	✓ (green)	✓	❌ (blue)	✓	❌ (blue)
make_heap	❌ (pink)	✓	❌ (pink)	✓	✓ (green)	✓ (green)	✓	❌ (blue)
BST find, insert, remove	❌ (pink)	✓	❌ (pink)	✓	✓ (green)	✓ (green)	✓	❌ (blue)
Sorting	❌ (pink)	✓	❌ (pink)	✓	❌ (pink)	✓ (green)	✓ (green)	? (red)

Big-Theta Expresses “Tight Bounds”

For f,g functions $\N \rightarrow \R^+$, $f(n) = \Theta(g(n))$ means there are $c_1, c_2,n_0 > 0 \text{ s.t. } n>n_0 \implies c_1\cdot g(n) \leq f(n) \leq c_2\cdot g(n)$

i.e. f asymptotically bounded from above and below by g

(Diagram with three lines, wobbly but roughly linear, with them stacked in the following order from top to bottom:

• $$c_2 \cdot g$$
• f
• $c_1 \cdot g$)

or f grows asymptotically the same as g.

“Grows the same as” is systemetic

Fact: $f(n) = \Theta(g(n)) \iff g(n) = \Theta(f(n))$

i.e. f grows the same as g $\iff$ g grows the same as f.

P.f.: $$f(n) = \Theta(g(n))

• $$\iff \exists \text{c1},\text{c2},n_0 > 0 \text{ s.t. } \forall n>n_0 \quad c1\cdot g(n) \leq f(n) \leq c2\cdot g(n)$$
• $$\iff \exists c_1,c2,n_0 > 0 \text{ s.t. } \forall n>n_0 \quad \frac{1}{c2} f(n) \leq g(n) \leq \frac{1}{c1} f(n)$$
• $$\iff \exists c_3,c_4,n_0 > 0 \text{ s.t. } \forall n>n_0 c_3 f(n) \leq g(n) \leq c_4 f(n)$$
• $$\iff g(n) = \Theta(f(n))$$

End