Questions tagged [master-theorem]

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

23 questions with no upvoted or accepted answers
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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) + \...
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0answers
32 views

Possible to use master method on T(n)/g(n)=aT(n/b)+f(n)

The master theorem can be used in case of a recurrence relation of the form 1) $T(n) = aT(\frac{n}{b}) + f(n)$ My question is whether it can be applied if 2) $\frac{T(n)}{g(n)} = aT(\frac{n}{b}) + ...
2
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0answers
37 views

Reference request: Leaf-heavy master theorem algorithms

I know many algorithms that can be analyzed using master theorem, but the only algorithm I know where the time is dominated by the leaves is fast matrix multiplication. Are there other recursive ...
1
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1answer
42 views

Using inductive hypothesis on recurrence relation?

I have a recurrence relation as follows $$T(n) = 2T(\lfloor n/2\rfloor) + n\log(n)$$ Using the induction hypothesis how do I obtain a relation $T(n)\leq E$ such that $E$ contains neither $T$ nor floor ...
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0answers
127 views

Advanced Master Theorem?

I have learned the Master's Theorem from the CLRS textbook (2nd Edition), the form of the Master Theorem given in the above text is associated with the proof of each and every case. So at the end I ...
1
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0answers
107 views

How to use master theorem to solve $T(n)=4T(n/8) + \sqrt n (\log_2 n)^2$

I want to solve the following using master theorem. $T(n)=4T(n/8) + \sqrt n (\log_2 n)^2$ I have: $a=4, b=8,f(n)=\sqrt n (\log_2 n)^2$ I calculate $n^{log_b a} = n^{\log_8 4} = n^{2/3}$ I ...
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0answers
132 views

How to prove a recursive's function Big-Theta without using repeated substitution, master theorem, or having the closed form?

I have a function defined: $V(j, k)$ where $j, k \in \mathbb{N}$ and $t > 0 \in \mathbb{N}$ and $1 \leq q \leq j - 1$. Note $\mathbb{N}$ includes $0$. $V(j, k) = \begin{cases} tj & k \leq 2 \\...
0
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1answer
47 views

What is the asymptotic bound for $T(n)= 3T(\sqrt[3]{n})+n^3$?

What is the asymptotic bound? How do you get to the result? $$T(n)= 3 \cdot T(\sqrt[3]{n})+n^3$$
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1answer
32 views

Divide and conquer recurrence relation

I have divide and conquer problem and below is the recurrence relation for it $$\begin{align}t (n) &= a\cdot t (n/4) + O (n^2/\log(n)) + O(n^2)\\ t(n) &= a\cdot t (n/4) + O(n^2) \end{align}$$ ...
0
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1answer
35 views

How to work out the odd case?

I am trying to solve this by using Substitution method. My solution must work both for even n-s and odd n-s. For evens case I have solved it. But for the odd's case I am stuck at this point. Hot to ...
0
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1answer
116 views

Solving the recurrence $T(n)=T(n-2)+n^2$ with the iterative method

I'm trying to solve this recurrence. I applied the iterative method: $$T(n) = T(n-2)+n^2$$ $$=T(n-4)+(n-2)^2+n^2$$ $$=T(n-6)+(n-4)^2+(n-2)^2+n^2$$ $$\cdot$$$$\cdot$$$$\cdot$$ $$=T(n-2k) + \sum_{i=0}^{...
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0answers
30 views

Solving a recurrence in which $n$ decreases by $\sqrt{2n}$

I'm trying to solve the recurrence $T(n)= 2T(n-\log f(n))+ f(n)$, where $f(n) = 2^{\sqrt{2n}}$, using the master theorem. Which case applies here?
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33 views

Iterative-substitution method yields different solution for T(n)=3T(n/8)+n than expected by using master theorem

I's like to guess the running time of recurrence $T(n)=3T(n/8)+n$ using iterative-substitution method. Using master theorem, I can verify the running time is $O(n).$ Using subtitution method however, ...
0
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2answers
129 views

Proving complexity of $T(n)=2T(n/3 + 1) + n$ non-Akra-Bazzi

We know that the complexity of $T(n)=2T(n/3 + 1) + n$ is $\Theta(n)$, as has been proved on this exchange before. However, what about proving it inductively? I believe that this method might work. ...
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33 views

Does the master theorem applies to this recurrence?

The recurrence: $T(n) = pT(n/q) + \log n$ for p < q and p >= 2. So, I've figured out it would fall into case 1, since we have $n^{log_{q}p} = n^r $, for $0<r<1$, which would mean that $f(...
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0answers
42 views

How to find running time complexity of divide and conquer method without Master Theorem

I understand that Master Theorem can be used to solve divide-and-conquer run times if they're in the form of $T(n) = aT(\frac{n}{b}) + n^clog^k(n)$ The reason behind it has to do with drawing a tree ...
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0answers
131 views

Master Theorem Help - How to make the value of “b” greater than 1 as required by the Master Theorem? What math is involved?

I know how to identify the parts of the Master Theorem, and I know that it is a recurrence relation. I don't understand how to make the "b" value greater than 1. Please see the image. I know the math ...
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0answers
106 views

Finding runtime of a recurrence relation with a fractional power

Consider the following algorithm and find the tightest Big-$O$: Assume $\texttt{multiplyKS}$($A,B$) is $O(n^{1.58})$ and $\texttt{Add}($A,B$)$ is $O(n)$. If my runtime is $T(n)$, I have: Lines 1 ...
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29 views

The applicability of the Master Theorem and calculation of asymptotic limits

Given the following recursive equation $T(n)=3T(\dfrac{n}{8})+ Θ(n^{1/3})$ I want to know how to explain the applicability of the Master theorem in a rigorous way and what means asymtotic limits of ...
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0answers
500 views

How to find lower bound of f(n) for master theorem

I'm studying the master method to solving a recurrence. It describes three cases, the last one of which depends on what lower bound a function f(n) has. I usually ...
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2answers
77 views

Using Iterative method to find recurrence relation vs Master Theorm

I'm trying to solve this recurrence relation using the iterative method and i keep getting the different answer from using the master theorem. $$\begin{aligned} T(n) &= 5T(n/2) +n^2 \\ &=...
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1answer
294 views

A difficult master theorem problem

Consider the function $B:\mathbb{N}\rightarrow\mathbb{R}$ defined by $$ B(n) = \begin{cases} 1 &\text{if $n\le 2$}\\ B\left(\left\lceil\frac{n}{\log_2n}\right\rceil \right)+n&...
-1
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1answer
170 views

Master Theorem Questions?

NOTE: I asked this on mathstackexchange, but didn't get the responses I wanted, thought I should post in CS. Sorry if i did something wrong but i am a newbie. State the asymptotic (worstcase) ...