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Intermediate Topic 23 of 51 Algorithms

Master Theorem

Formula solving recurrences of divide-and-conquer algorithms.

How it works

For T(n) = aT(n/b) + f(n), compare f(n) with n^(log_b a): if f is smaller the leaves dominate, if equal you get an extra log n factor, if larger the root dominates (with a regularity condition).

Where it is used

Quickly solving merge-sort style recurrences: T(n)=2T(n/2)+n gives Θ(n log n).

Common pitfall

Does not apply to unequal splits or non-polynomial gaps.

Where it fits in the course

Master Theorem is topic 23 of 51 in Algorithms, rated Intermediate. It follows Divide and Conquer. The next topic is Greedy Algorithms.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Algorithms quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Master Theorem?
Formula solving recurrences of divide-and-conquer algorithms. It belongs to Algorithms and is an intermediate-level topic.
Where do people use Master Theorem?
Quickly solving merge-sort style recurrences: T(n)=2T(n/2)+n gives Θ(n log n).
What is the most common mistake with Master Theorem?
Does not apply to unequal splits or non-polynomial gaps.
What should I learn before this topic?
Start with Divide and Conquer, the topic just before it in Algorithms.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Algorithms quiz. You can also explore Growth Rates (Big-O) in the Visual Lab.
Is Master Theorem hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Master Theorem?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Master Theorem?
Divide and Conquer, Greedy Algorithms, Timsort are closely connected. The full list is in the related topics section of this page.
What should I study after Master Theorem?
Continue with Greedy Algorithms, the next topic in Algorithms.

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