Math Is Fun Forum

azair

Member

Registered: 2013-01-08

Posts: 5

Help!!! Master Theorem?

Hello everyone , I am learning an Algorithm analysis on my own and today I came across 'Master Theorem for Divide and Conquer'. Since I'm quite not good at Mathematics, this topic is giving me a full headache.(ahem ahem, no offense please!!! ;-)).
Alright, The definition is given as follows :

"If the recurrence is of the form T(n)=aT(n/b)+Θ(n^k log^p n),where a>=1, b>1, k>=0 and p is a real number, then:
1.)  If a>b^k, then T(n)=Θ(n^log^a↓b)   [Note : lets assume ↓ as base.]

2.)  If a=b^k :
               a.) If p>-1, then T(n)=Θ(n^log^a↓b * log^p+1 n)
               b.) If p=-1, then T(n)=Θ(n^log^a↓b * loglog n)
               c.) If p<-1, then T(n)=Θ(n^log^a↓b).

3.) If a<b^k :
               a.) If p>=0, then T(n)=Θ(n^k log^p n)
               b.) If p<0, then T(n)=O(n^k).

"
Well can anyone help me explain what these means in simple terms with an example if possible.
For example : Problem--> T(n)=2T(n/2)+nlogn. [The Answer is Θ(nlog logn) :? How???]

{I'm assuming this tutorial's topic as : MASTER THEOREM FOR DUMMIES. }
Thanking you in advance...

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anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Help!!! Master Theorem?

What do you mean by log^p?

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,634

Re: Help!!! Master Theorem?

I think it is

What I don't follow is the theta function notation.  Do you recognise that?

Bob

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Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Help!!! Master Theorem?

I do. It means that the function T(n) grows the same way as the function isnide the theta...

I think somethong is wrong either with tthe logarithms or with the conditions for p in the second case (a=b^k)...

Last edited by anonimnystefy (2013-01-10 00:43:12)

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.