Lecture 05 - Jan 13

Asymptotic Notations

Big O Notation - Upper Bound

if constants &

such that

Big Ω - Lower Bound

if constants ,

such that

Big Θ - Tight Bound

if constants , &

such that

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Examples

Example 1: Proving Big O

Given:

Show that :

for all

Therefore, with and

Example 1b: Sum Formula

Given:

Show that :

Therefore, with and

Since for , we also have

Thus,

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Example 2: Recurrence Relation

Given:

Solving by substitution:

After iterations:

When , we have , so

Substituting:

Therefore:

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Example 3: Additional Recurrence Relations

Given:

This is since for

Recurrence 1:

By Master Theorem:

Recurrence 2:

By Master Theorem: