What is the Master Theorem?

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En provenance du cours de University of California San Diego

Algorithmic Toolbox

3675 notes

University of California San Diego

Algorithmic Toolbox

3675 notes

Cours 1 sur 6 dans Specialization Data Structures and Algorithms

The course covers basic algorithmic techniques and ideas for computational problems arising frequently in practical applications: sorting and searching, divide and conquer, greedy algorithms, dynamic programming. We will learn a lot of theory: how to sort data and how it helps for searching; how to break a large problem into pieces and solve them recursively; when it makes sense to proceed greedily; how dynamic programming is used in genomic studies. You will practice solving computational problems, designing new algorithms, and implementing solutions efficiently (so that they run in less than a second).

À partir de la leçon

Divide-and-Conquer

In this module you will learn about a powerful algorithmic technique called Divide and Conquer. Based on this technique, you will see how to search huge databases millions of times faster than using naïve linear search. You will even learn that the standard way to multiply numbers (that you learned in the grade school) is far from the being the fastest! We will then apply the divide-and-conquer technique to design two efficient algorithms (merge sort and quick sort) for sorting huge lists, a problem that finds many applications in practice. Finally, we will show that these two algorithms are optimal, that is, no algorithm can sort faster!

What is the Master Theorem?4:55

Proof of the Master Theorem9:45

Rencontrer les enseignants

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering
Pavel Pevzner
Professor
Department of Computer Science and Engineering
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics

Here we're going to talk about the master theorem.

We'll describe what the master theorem is and how to use it.

And we'll reserve to the next video a proof.

So we've had many occasions where we have had to write a recurrence relation for

a divide and conquer problem.

This is an example of one for binary search.

We break a problem down into a problem half as big and

we do a constant amount of work at each level.

And this gives us a solution T(n) = O(log n).

The problem is for each one of these we have to create a recurrence tree,

figure out how much work is done at each level, sum up that work.

That's a lot to do to solve each recurrence relation.

Here's an example that we used for the polynomial multiplication.

So we broke a problem into four sub-problems, each half the size,

and did a linear amount of work.

And the solution was T(n) = O(n squared).

When we had the more efficient algorithm, where we had only three sub-problems

instead of four, we then got a solution of O(n to the log base 2 of 3).

Sometimes we break a problem into only two subproblems and

there the solution is O(n log n).

So, wouldn't it be nice if there was a way that we just had a formula to

tell us what the solution is rather than having to create this recurrence tree each time?

And that's what the Master Theorem basically does.

So, the Master Theorem says if you have a recurrence relation T(n) equals a, some

constant, times T( the ceiling of n divided by b)

+ a polynomial in n with degree d.

And that ceiling, by the way, could just as well be a floor or

not be there at all if n were a power of b.

In any case, the a is a constant greater than 0.

b is greater than 1 because we want to actually make sure the problem size

gets smaller.

And d is greater than equal to 0.

Well, in that case, we have a solution for T of n.

There are three sub cases.

Case number 1, and all of these cases depend on the relationship between d,

a, and b.

In particular, is d greater than log base b of a?

If so, the solution is just this polynomial in n, O(of n to the d).

If d is exactly equal log base b of a,

then the solution is big O of n to the d with an extra factor of log n.

And finally, if d is less than log base b of a,

then the solution is big O of n to the log base b of a.

So let's look at some applications of this theorem.

So here's one where we go back to the polynomial multiplication.

Here a is 4, b is 2, and d is 1.

Because O(n) is just O(n to the 1).

And we look at the relationship between d, which is 1, and

log base b of a, which is log base 2 of 4 or 2.

Well clearly d is less than log base b of a, so we're in case three.

Therefore T(n) = O(n to the log base b of a),

or just O(n squared).

If now we change the 4 to a 3, a is 3, b is 2, d is 1.

Now d is still less than log base b of a because log

base 2 of 3 is greater than 1, and so again we're in case three.

T(n) equals O(n to the log base b of a), which equals O(n to the log base 2 of 3).

If we reduce the 3 down to a 2 what happens?

Well here, a is 2, b is 2, d is 1.

Log base b of a is log base 2 of 2, which is just 1.

So now d is equal log base b or a.

We're in case two now.

And so, T of n equals O(n log n).

And now this shows an example also of case two.

So this is the binary search example.

A is 1, b is 2, d is 0.

Well the log base two of one, log base b of a, is equal to zero.

So d is equal to log base b of a.

We're in case two, T(n) = O(n to the d log n),

which is in the 0 log n, which is just O(log n).

And a final example where we are actually in case one.

So here a is 2, b is 2, and d is 2.

So log base b of a is log base 2 of 2, which is one.

So d is now greater than log base b of a.

We are now in case one, T(n) equals O(n to the d), which is O(n squared).

So what we've seen now is that we have this master theorem that allows us,

for most recurrences, when you do a divide and conquer

which fit into this general formula, allows us to easily figure out

which case we are based on the relationships between a, b, and d.

And then figure out the result quite quickly.

In our next video we'll look at a proof of why the master theorem works.

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