Math Is Fun Forum

dannyv

Member

Registered: 2007-09-20

Posts: 34

Divide and Conquer Algorithms

Does anyone read the book "Algorithms" by Papadimitriou et al.? I found it to be an excellent book, just sometime need a little bit of explanation in some parts of it. For example, in chapter 2 about divide and conquer algorithms it presents a master theorem about the time complexity for this kind of algoritms. It saids something like this:

"If T(n)=aT(n/b)+O(n^d) for some constants a>0, b>1 and d>=0 then

T(n)= O(n^d)               if d>log_b(a)
T(n)= O(n^d log n)       if d=log_b(a)
T(n)= O(n^{log_b(a)}) if d<log_b(a)

Where n is the size of the input, a is the branching factor and b is how much we partition the input in each recursion step."

Can anyone tell me what is the parameter d?

Thanks

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Divide and Conquer Algorithms

When you have a divide and conquer algorithm, they do something on each step after they divide.  O(n^d) is the cost of what they do on this step.  So let's say my imaginary algorithm divides a list in half, then sorts each part of that list with bubble sort.  The cost of the bubble sort is n^2, so d = 2.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: Divide and Conquer Algorithms

Thanks, know is cristal clear!! :-)