Questions on the Master theorem, a method for obtaining asymptotic bounds on recurrences of a specific form.

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2answers
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$T(n)=2T(n/2)+n\log n$ and the Master theorem [duplicate]

According to Introduction to algorithms by Cormen et al, $$T(n)=2T(n/2)+n\log n$$ is not case 3 of Master Theorem. Can someone explain me why? And which case of master theorem is it?
3
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4answers
53 views

Asymptotic Runtime of Interrelated Functions

I have two functions $S$ and $T$ which are interrelated and I want to find the asymptotic worst case runtime. The fact that they are interrelated is stumping me... How would I find the asymptotic ...
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2answers
107 views

Solution to recurrence $T(n) = T(n/2) + n^2$

I am getting confused with the solution to this recurrence - $T(n) = T(n/2) + n^2$ Recursion tree - ...
0
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1answer
69 views

Master theorem for $T(n) = 2T(n/2) + n^{2}\log n$

Would I use the third case of the Master Theorem for the recurrence equation $T(n) = 2T(n/2) + n^{2}\log n$? The condition given for the third case by Wikipedia is $f(n) = \Theta(n^c)$ when $c > ...
4
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0answers
49 views

Are there master theorems that deal with parameters of the form $n-c$?

While thinking about this question on a recurrence I checked out some stronger master theorems. Unfortunately, they do not seem to apply because terms $\qquad\displaystyle T(n) = \dots + T(n-1) + ...
6
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1answer
67 views

Why does Akra-Bazzi need that toll-function g is bounded?

Following up on vonbrand's answer I want to write a small document about stronger master theorems for our students, one of which is the Akra-Bazzi theorem. I have copied the theorem from their paper ...
4
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1answer
369 views

Recursive equation for complexity: T(n) = log(n) * T(log(n)) + n

For analyzing the running time of an algorithm , I'm stuck with this recursive equation : $$ T(n) = \log(n) \cdot T(\log n) + n $$ Obviously this can't be handled with the use of the Master Theorem, ...
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1answer
161 views

Solving a recurrence with the Master Theorem

Problem taken from here (page 3): http://cse.unl.edu/~choueiry/S06-235/files/MasterTheorem-Handout.pdf $T(n) = 3T(\frac{n}{2}) + \frac{3}{4}n + 1$ $f(n) = \frac{3}{4}n + 1$ It says we cannot use ...
4
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1answer
266 views

Finding recurrence when Master Theorem fails

Following method is explained by my senior. I want to know whether I can use it in all cases or not. When I solve it manually, I come to same answer. $T(n)= 4T(n/2) + \frac{n^2}{\lg n}$ In above ...
2
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1answer
81 views

Not sure if my solution to following recurrence is correct

I have a recurrence relation, it is like the following: $T(e^n) = 2(T(e^{n-1})) + e^n$, where $e$ is the base of the natural logarithm. To solve this and find a $\Theta$ bound, I tried the ...
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0answers
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The use of master theorem appriopriately [duplicate]

I have a recurrence relation and trying to use master theorem to solve it. The recurrence relation is: $T(n) = 3T(n/5) + n^{0.5}$ Can I use the master theorem in that relation? If so, can I say that ...
3
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2answers
142 views

Usage of master theorem for solving recursions

I know that master theorem is used for the recurrence relations of the form: T(n) = aT(n/b) + f(n) But in my question, i am supposed to solve the following recurrence relation by using master ...
11
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1answer
249 views

Rigorous proof for validity of assumption $n=b^k$ when using the Master theorem

The Master theorem is a beautiful tool for solving certain kinds of recurrences. However, we often gloss over an integral part when applying it. For example, during the analysis of Mergesort we ...
3
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2answers
460 views

How to the examples for using the master theorem in Cormen work?

I'm reading Cormen's Introduction to Algorithms 3rd edition, and in examples of Master Method recursion solving Cormen gives two examples $3T( \frac{n}{4} ) + n\log(n)$ $2T( \frac{n}{2} ) + ...
0
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1answer
130 views

Explanation of a specific recurrence with respect to Master Theorem

Concerning the Master Theorem. I have found the following equation as the base of analysis: $\quad T(n) = aT(n/b) + \Theta(n^k)$ but I also found the following: $\quad T(n) = aT(n/b) + ...
6
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3answers
921 views

Solving a recurrence relation with $\sqrt{n}$ as parameter

Let $T(n) = \sqrt{n} T(\sqrt{n}) + c\,n$ for $n \gt 2$ and some positive constant $c$ and $T(2) = 1$. I know the Master theorem, but I'm not sure as to how we could solve this relation using it. Any ...
0
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1answer
199 views

Big Omega of $n \log n$

While studying master method at recurrences topic I'm stacked at a point. It is written in the book as: $T(n) = 3T(n/4) + n \log n$, we have $a = 3, b = 4$, $f(n) = n \log n$, and ...
4
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2answers
236 views

Problems showing the constraint of master theorem case three holds

Prove or disprove the following statements: $T\left( n \right) = 2T\left( {\frac{n}{2}} \right) + f\left( n \right),f\left( n \right) = \theta \left( {{n^2}} \right) $ then $ {\rm{ }}T\left( n ...
7
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2answers
2k views

Why is there the regularity condition in the master theorem?

I have been reading Introduction to Algorithms by Cormen et al. and I'm reading the statement of the Master theorem starting on page 73. In case 3 there is also a regularity condition that needs to be ...
3
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1answer
603 views

Solving $T(n)= 3T(\frac{n}{4}) + n\cdot \lg(n)$ using the master theorem

Introduction to Algorithms, 3rd edition (p.95) has an example of how to solve the recurrence $$\displaystyle T(n)= 3T\left(\frac{n}{4}\right) + n\cdot \log(n)$$ by applying the Master Theorem. I am ...
5
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1answer
147 views

Finding lambda of Master Theorem

Suppose I have a recurrence like $T(n)=2T(n/4)+\log(n)$ with $a=2, b=4$ and $f(n)=\log(n)$. That should be case 1 of the Master theorem because $n^{1/2}>\log(n)$. There is also a lambda in case 1: ...
7
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2answers
961 views

Master theorem not applicable?

Given the following recursive equation $$ T(n) = 2T\left(\frac{n}{2}\right)+n\log n$$ we want to apply the Master theorem and note that $$ n^{\log_2(2)} = n.$$ Now we check the first two cases for ...
11
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3answers
531 views

Solving Recurrence Equations containing two Recursion Calls

I am trying to find a $\Theta$ bound for the following recurrence equation: $$ T(n) = 2 T(n/2) + T(n/3) + 2n^2+ 5n + 42 $$ I figure Master Theorem is inappropriate due to differing amount of ...
4
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1answer
221 views

Master theorem and constants independent of $n$

I applied the Master theorem to a recurrence for a running time I encountered (this is a simplified version): $$T(n)=4T(n/2)+O(r)$$ $r$ is independent of $n$. Case 1 of the Master theorem applies ...
2
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1answer
531 views

Solve a recurrence using the master theorem

This is the recursive formula for which I'm trying to find an asymptotic closed form by the master theorem: $$T(n)=9T(n/27)+(n \cdot \lg(n))^{1/2}$$ I started with $a=9,b=27$ and $f(n)=(n\cdot \lg ...