People who code: we want your input. Take the Survey

Questions tagged [master-theorem]

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

Filter by
Sorted by
Tagged with
20
votes
1answer
903 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 ...
18
votes
5answers
20k views

Solving a recurrence relation with √n as parameter

Consider the recurrence $\qquad\displaystyle T(n) = \sqrt{n} \cdot T\bigl(\sqrt{n}\bigr) + c\,n$ for $n \gt 2$ with some positive constant $c$, and $T(2) = 1$. I know the Master theorem for ...
16
votes
2answers
17k 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 ...
15
votes
3answers
8k 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 ...
11
votes
2answers
18k 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 $...
9
votes
1answer
1k views

Intuition behind the Master Theorem

The Master Theorem provides a method of solving recurrence relations for divide-and-conquer algorithms. It was first presented to me in my intro algorithms class as the following: For a recurrence ...
8
votes
1answer
17k 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 ...
8
votes
2answers
977 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 [...
6
votes
1answer
818 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 ...
6
votes
1answer
2k views

Applying the Master Theorem on Merge sort

I found the proof below in a textbook. I would like to know why it is important for the proof that it uses $\lceil \frac{n}{2} \rceil$ instead of just $\frac{n}{2}$? I know that you can't split into ...
6
votes
1answer
175 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: ...
6
votes
0answers
205 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) + \...
5
votes
2answers
19k views

$T(n)=2T(n/2)+n\log n$ 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?
5
votes
2answers
636 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 \right)...
5
votes
1answer
2k 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 ...
5
votes
1answer
2k views

Cases of Master Theorem

Suppose that we have $ \\ T(n)=\left\{\begin{matrix} c, & \ \text{if } n<d\\ aT\left( \frac{n}{b} \right )+f(n), & \ \ \text{if } n \geq d \end{matrix}\right.$ The Master theorem is the ...
4
votes
2answers
459 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 ...
4
votes
1answer
557 views

Master Method to solve recurrences is 'a' related to 'b'?

The master method allows us to solve certain recurrences of the form $$T(n) = aT(n/b)+f(n)\,,$$ where $a\ge1$ and $b>1$ are constants and $f(n)$ is a positive function with some further ...
4
votes
1answer
626 views

Master Theorem and rounding up to the nearest integer

For the master theorem for recurrences of the form $$T(n) = a\,T\!\left(\tfrac{n}{b}\right) + f(n)\,,$$ what difference would it make if the split was into calls of $\lceil n/b\rceil$ instead of $n/...
4
votes
3answers
8k views

Solving recurrence relation with square root

I am trying to solve the following recurrence relation :- $T(n) = T(\sqrt{n}) + n$ using masters theorem. We can substitute $n = 2 ^ m$ $T(2^m) = T(2 ^ {\frac{m}{2}}) + 2^m$ Now we can rewrite it ...
4
votes
1answer
2k views

What is the case 2 in master theorem?

I am confused about the statement of the Master theorem in CLRS book. Here is the link of the book CLRS. In page 94, the theorem, in case 2, states that: If $\displaystyle f(n)=\Theta(n^{\log_ba})$,...
4
votes
1answer
2k 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, ...
4
votes
5answers
15k views

Solving T(n) = 3T(n/3)+n/2 using master method

I thought I understood the Master Method quite well till I saw this question $T(n) = 3T(\frac{n}{3})+\frac{n}{2}$ My approach: $a = 3 ; b=3$ and $f(n) = \frac{n}{2}$ $n^{\log_b{a}}$ = $n^{log_3{...
3
votes
5answers
19k views

How to solve T(n)=2T(√n)+log n with the master theorem?

I'm trying to solve the recurrence $$T(n)=2T(\sqrt{n})+\log n$$ using the master theorem. Which case applies here?
3
votes
4answers
78 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 ...
3
votes
2answers
18k 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 - ...
3
votes
3answers
2k views

Master Theorem: How to find the value of b in this recurrence relation

The master theorem is used with recurrences of the form T(n) = aT(n/b) + f(n) where a >=1 and b > 1, in which case the value of b can be easily seen from the ...
3
votes
2answers
5k views

How do we derive the runtime cost of Karatsuba's algorithm?

I've read the Wikipedia article explaining the complexity analysis of the Karatsuba algorithm, but I'm not fully grasping it. I seem to have gotten about 75% of the way to the solution on my own, but ...
3
votes
1answer
2k 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 n)^...
3
votes
1answer
62 views

Justifying a claim in the proof of the master theorem

I am trying to understand the proof of the master theorem and I came up with my own proof for why (4.23) is true. My argument is as follows: Claim: $g(n)=O\left(\sum_{i=0}^{\log_{b}(n)-1}a^i(n/b^i)^{\...
3
votes
2answers
209 views

$f(n) = o(n^c) \rightarrow \exists \epsilon > 0 \ s.t. f(n) = O(n^{c-\epsilon})$

I'm trying to prove that for arbitrary $c > 0$, $f(n) = o(n^c) \rightarrow \exists \epsilon > 0 \ s.t. f(n) = O(n^{c-\epsilon})$ Intuitively, this seems to be true to me (little-o implies ...
3
votes
2answers
2k 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 $3T( \frac{n}{4} ) + n\log(n)$ $2T( \frac{n}{2} ) + n\log(n)$ ...
3
votes
1answer
2k views

Meaning of polynomially larger or smaller in the context of the master method

I'm studying the master method of solving recurrences and I have a somewhat decent math background but I'm having difficulty understanding the concept of $n^{\log_ba}$ being polynomially smaller or ...
3
votes
1answer
171 views

Find the asymptotic bound $\Theta$ of $t(n)=t(\frac{n}{5})+t(\frac{n}{17})+n$

Find the asymptotic bound in terms of $\Theta$ (Theta) using the master theorem for the following recursive equation. Assume that $t(n)= \Theta (1)$ for suffucuently small $n$. $$t(n)=t(\frac{n}{...
3
votes
1answer
350 views

Can anyone explain why this is an inadmissible recurrence case that cannot be solved by the master theorem?

Wikipedia says that the following recurrence is inadmissible since there is a non-polynomial difference between $f(n) = \frac{n}{\log n}$ and $n^{\log_b a}$: $$ T(n) = 2T\left(\frac{n}{2}\right) + \...
3
votes
1answer
45 views

Is there a difference between using $n$ and $\Theta(n)$ in recurrences?

Is there a difference between $T(n)=2T(n/2)+n$ and $T(n)=2T(n/2)+Θ(n)$ when using the master theorem? I've seen it both ways and am a little confused. (Looking for the answer $nlogn$).
3
votes
1answer
158 views

Master Theorem on oscillating function

Consider a recurrence of the form $T(n) = a T(n/b) + f(n)$ Master theorem's regularity condition excludes some cases (for example, when $f(n)$ is oscillating). Suppose, however, that $f(n)$ is ...
3
votes
1answer
159 views

Applying Case 3 of Master Theorem to $T(n) = 9T(n/3) + n^3$

Given $T(n) = 9T(n/3) + n^3$, I know that $a =9$, $b=3$, and $f(n) = n^3$ and $n^{\log_{3}9} = n^2$ thus Case 3 applies: $n^{\log_{b}a} < f(n)$, $n^2 < n^3$. Can someone explain how to apply the ...
3
votes
1answer
149 views

Compare two complexity functions having the same asymptotic complexity

For a certain problem two solution algorithms (A1 and A2) with the following execution times have been found: $A1: T_{A1}(n)=4n^2 +7log(n^2)$ $A2: T_{A2}(n) = 4T(n/2) + log(n)$ Say, technically ...
2
votes
1answer
734 views

How to solve $T(n)= 4T(\sqrt n) +\log^2n$?

Consider the recurrence $$T(n)= 4T(\sqrt n) + \log^2n. $$ I am not able to solve this recurrence, since it involves a square root. Please help me with the solution.
2
votes
1answer
3k views

Master theorem for $T(n)=T(n-1)+O(n)$

The recurrence of selection sort is $$T(n) = T(n-1)+ O(n).$$ Can we apply the master theorem to this recurrence? I am confused because the master theorem can be applied to the following recurrence $$...
2
votes
2answers
6k views

What is the recurrence form of Bubble-Sort

I understand how bubble sort works and why it is O(n^2) conceptually but I would like to do a proof of this for a paper using the master theorem. As an example: The recurrence form for merge sort is ...
2
votes
3answers
297 views

Asymptotic equivalent of the recurrence T(n)=3⋅T(n/2)+n

The questions is to find the running time $T(n)$ of the following function: $$T(n)=3\cdot T(n/2) + n \tag{1}$$ I know how to solve it using Master theorem for Divide and Conquer but I am trying to ...
2
votes
1answer
321 views

Conditions for applying Case 3 of Master theorem

In Introduction to Algorithms, Lemma 4.4 of the proof of the master theorem goes like this. $a\geq1$, $b>1$, $f$ is a nonnegative function defined on exact powers of b. The recurrence relation for $...
2
votes
2answers
352 views

Formulating the master theorem with Little-O- and Little-Omega notation

In a lecture of Algorithms of Data Structures (based on Cormen et al.), we defined the master theorem like this: Let $a \geq 1$ and $b \gt 1$ be constants, and let $T : \mathbb{N} \rightarrow \...
2
votes
1answer
59 views

Could I apply the master theorem if my $N/b$ is $\varphi(N)$?

Let $$T(N) = \begin{cases}1 & \text{if } N = 1\\ T(\varphi(N)) + \lg(\varphi(N))^3 & \text{otherwise} \end{cases}$$ where $\varphi(N)$ is Euler's totient function. Can I somehow express $\...
2
votes
2answers
97 views

How to use Master Theorem with strange format of $b$ parameter?

I have a funcion $T: \mathbb{N}\to\mathbb{N}$ defined as: $$T(n)=\begin{cases} 6 &\text{ if } n=0,\\ T(n-1) + 6n + 6 &\text{otherwise.} \end{cases}$$ How can I apply the Master Theorem to ...
2
votes
1answer
81 views

Run time of a Simple Recurrence

Given the recurrence $T(n) = T(\sqrt{n}) + \theta(lglgn)$, provide an asymptotically tight bound on it's run time. My solution was to let $m = 2\sqrt{n}$, which leads to the recurrence $S(m) = S(m/2) ...
2
votes
2answers
89 views

Master theorem: what to do with constant in parenthesis?

In analysis of algorithms, we sometimes use the (unsimplified) Master Theorem for recurrence relations. What should be done in the case that there is a constant factor in the numerator following T? $$ ...
2
votes
1answer
51 views

Recurrence : $T(n) = 4T(n/2) + Θ(n^2/\log n)$

Is there a way to solve this recurrence using master theorem: $$T(n) = 4T(n/2) + Θ(n^2/\log n)$$