Lecture 06 - Jan 15

Master Theorem

Given: Let ,

Case 1

If for

Case 2

If , then:

Case 3

If for , and for some and sufficiently large (regularity condition), then:

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Simplified

If for constants , then

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Examples

Example 1

Solution:

• , ,

Compare with :

• for

Check regularity condition:

for ✓

By Case 3:

Example 2

Solution:

• , ,

Since

By Case 2:

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Substitution Method

Example 1: Solve using Substitution:

Solution:

Guess:

Prove by induction that for some :

if and is sufficiently large.

Therefore,

Example 2: Solve using Substitution:

Solution:

Let , so :

Let :

By Master Theorem (Case 2):

Substituting back:

Therefore,