Questions tagged [master-theorem]

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

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How to solve $T(n) = 27T(n/9) + n^3$ with substitution method

I'm trying to bound this recurrence with the substitution method. My guess is $O(n^3)$. These are some steps: $$T(n) \leq cn^3 \\ T(n) \leq 27cn^3+n^3$$ How can I continue?
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2answers
44 views

Regularity condition for cases 1 & 2

My question concerns the version of the Master Theorem described in CLRS and in this handout. I already understand the following: If the regularity condition in case 3 does not hold, then we can't ...
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1answer
43 views

Why is the time complexity of merge sort with a $\Theta(n^2)$ merge function $\Theta(n^2)$?

The original problem I was solving was what would the time complexity of a merge sort algorithm be, if it used a merge algorithm with complexity $\Theta(n^2)$ instead of $\Theta(n)$. The solution says ...
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1answer
51 views

How to solve $T(n) = 2T(n/4) + n \log n$ with substitution method?

I am trying to solve this recurrence with substitution method. I guess $T(n) = \Theta(n \log n)$ (with Master Theoreme). Can someone show me how to demonstrate the upper bound $T(n) = O(n \log n)$?
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1answer
37 views

Why doesn't master theorem solve $T(n) = 2T(n/2) + n\lg\lg n$?

Given two recurrences: $T(n) = 2 T(n/2) + n \lg \lg n$ $T(n) = 4 T(n/2) + n \lg \lg n$ I'd think that both works for master theorem, but the solution is that the first one cannot use masters ...
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1answer
30 views

How can we get upper bound in terms of Big Oh notation using Master theorem?

The recursion is: T(n) = 5T(n/2) + O(n) I solved for the time complexity using Master theorem and found Θ(n^2). but, the question has asked to find the upper bound ...
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1answer
116 views

Master Method: Divide and Conquer

According to my evaluation ,the overall asymptotic running time of the below algorithm is O(n) ,since x (number of recursive ...
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2answers
35 views

recurrence with exponentials

I am trying to figure out on how to approach the problem on finding proving the asymptotic of an exponential recurrence. It is described as such: t(n)=4t(n/2)+2^n with t(1)=1 for n>=5 From what I ...
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0answers
64 views

Total work done at a recursion tree level

In the proof of Master theorem in Dasgupta's Algorithms the author says that the total work done at a recursion tree level is $$a^k \times O\left(\frac{n}{b^k}\right)^d$$ where $a$ is the branching ...
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2answers
92 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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3answers
153 views

How to solve $T(n)=4T(\sqrt{n}/3)+(\log n)^2$ with the master theorem?

Can somebody help me with this recurrence please? $T(n)=4T(\sqrt{n}/3)+(\log n)^2$
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2answers
105 views

Solving constants in the recursive term with master theorem

We are learning how to solve recurrence relations in different ways (Forward Substitution, Backward Substitution, Master Theorem, etc...). I really thought I understood the topic since most of the ...
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1answer
50 views

Divide and conquer recurrence relation

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

Master's Theorem recurrence

Given recurrence relation $T(n)=8T(n/6)+n \log n$, I get that the running time of the leaves should be $n^{\log_6 8}$ and $f(n)$ should be $n \log n$, but how can I know which one is bigger ?
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1answer
41 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 ...
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3answers
51 views

Is $n \log n$ in $O(n^{1.46-\varepsilon})$?

I am trying to figure out the solution of the recurrence relation $$T(n) = 5T(n/3) + n \log n$$ using the Master Method. I am guessing that $f(n) = O(n^{1.46 - \varepsilon})$, but I am confused in the ...
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1answer
73 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)^{\...
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1answer
50 views

Master theorem: $T(n)=10T(n/9)+n\lg(n)$

I am told to solve the recurrence $$T(n)=10T(n/9)+n\lg(n)$$ using the Master theorem. I then try to use case 3. However, I am unable to show that for $f(n)=n\lg(n)$ then $10f(n/9) \leq cn\lg(n)$ for $...
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1answer
100 views

Does Master Theorem apply to $T(n) = 4T(n/2) + n^2 \log n$

Based on CLRS Theorem 4.1, master theorem doesn't apply to $T(n) = 4T(n/2) + n^2 \log n$. However, I saw the 4th condition of master theorem on slides of Bourke. If $f(n)=\Theta(n^{\log_ba}\log^kn)$, ...
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1answer
56 views

Recurrence relations and the Master Theorem

Although it might be a bit of newbie question, my question is, How can I apply the Master theorem to the following relation: T(n) = 99T(n/100) + log(n!) I'm trying ...
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1answer
367 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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1answer
52 views

How to prove $T(n) = 2T(n/2) + n/\log(n)$ can't be solved using the Master Theorem?

I have read (in this question) that this recursion can't be solved via Master Theorem. But I couldn't find exact and complete proof why the Master Theorem does not apply.
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1answer
889 views

Solving $T(n) = 4T(n/2) + n^3$ with substituton method

I am trying to solve the following recurrence $T(n) = 4T(n/2) + n^3$ with substitution method. My guess is $T(n) = \Theta (n^3)$ (I used master theorem) and I tried to show that $T(n) \leq cn^3$. But, ...
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1answer
63 views

Intuition on O(number of leaves) for master theorem

I am trying to develop the intuition of the master theorem for the case where $a > b^{d}$ [Case 3] in this video. In the video, they say that since most of the work is done at the leaves, we should ...
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2answers
63 views

Solving $T(n)=3T\bigl(\bigl\lfloor \frac{n}{3}\bigr\rfloor\bigr) +2n\log n$ without the Master Theorem

I want to solve $$T(n)=3T\bigl(\bigl\lfloor \frac{n}{3}\bigr\rfloor\bigr) +2n\log n,$$ with base case $T(n) = 1$ if $n \leq 1$. I know that the solution is(with the help of the Master Theorem) $$\...
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1answer
42 views

Solving $T(n)=3T(\lfloor \frac{n}{3}\rfloor) +2n\log n$

I want to solve $$T(n)=3T\bigl(\bigl\lfloor \frac{n}{3}\bigr\rfloor\bigr) +2n\log n,$$ with base case $T(n) = 1$ if $n \leq 1$. I am sure that the Master Theorem does not work. I am trying a lot with ...
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2answers
151 views

Solving T(n) = 2*T(n-1)+4 witht the Master Theorem

I am wondering if there is a way to solve a recurrence time function with the master theorem if no $b$ exists. Like in this case. $$ T(n) = 2\times T(n-1)+4$$
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1answer
636 views

Solving $T(n) = 16T(n/2) + n$

I am trying to solve the following recurrence relation :- $T(n)=16T(n/2)+n$ using masters theorem. I got $\Theta (n^2)$ (Which matched the first case in the theory) which is wrong, any help with this ...
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0answers
31 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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1answer
63 views

Solution to T(n) = 2T(n/2) + log n

So my recursive equation is T(n) = 2T(n/2) + log n I used the master theorem and I find that a = 2, b =2 and d = 1. which is case 2. So the solution should be O(n^1 log n) which is O(n log n) I looked ...
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1answer
135 views

Solving T(n) = 3T(n/3)+sqrt(n) in terms of 𝜃 or O notions using master method [duplicate]

Please help me solve it in terms of Theta or Big O
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1answer
1k views

Solving $T(n) = 3T(\frac{n}{3})+\sqrt{n}$ using master method

How can I use the master's method in order to solve the recurrence formula $T(n)=3T(\frac{n}{3})+\sqrt{n}$ ?
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2answers
125 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? $$ ...
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1answer
568 views

Solve Recurrence for $T(n) = 7T(n/7) + n$

I'm trying to solve the recurrence for $T(n) = 7T(n/7) + n$. I know using Master Theorem it's $O(n\log_7n)$, but I want to solve it by substitution method. At level $i$, I get: $7^i T(n/7^i) + (n+7n+7^...
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1answer
706 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 ...
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1answer
54 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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2answers
234 views

solving recurrence T(n) = T(√n) + theta((lg lg n))

My solution: let m = lg n. Then n = 2^m. T(2^m) = T(2^(m/2)) + theta(lgm). Let S(m) = T(2^m). Then S(m) = S(m/2) + theta(lgm). Applying master theorem, I get m^(lg1) = 1 which is asymptotically ...
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1answer
71 views

Does $20n$ belong to $O(n^{1-\epsilon})$ for some $\epsilon > 0$?

I am quite new to master theorem and I would like to ask the following question for $$𝑇(𝑛)=4𝑇(𝑛/4)+20𝑛.$$ If there is a constant value like $20n$ does it affect the equation? Would the equation ...
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1answer
60 views

Proof of inequality $\lceil x \rceil \le x+1$

I went through the Master Theorum extension for floors and ceiling section 4.6.2 in the book Introduction to Algorithms It had the following statement: Using the inequality $\lceil x \rceil \le x+1$ ...
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1answer
189 views

Clarification of the proof involving the regularity condition in Master Theorem

I was going the text Introduction to Algorithms by Cormen et al. Where I came across the following statement in the proof of the third case of the Master's Theorem. (The Statement of Master theorem) ...
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0answers
182 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 ...
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2answers
60 views

Master Theorem applicable here?

Let $T(n):=\begin{cases} \frac{2+\log n}{1+\text{log}n}t(\lfloor\frac{n}{2}\rfloor) + \log ((n!)^{\log n}) & \text{if }n>1 \\ 1 & \text{if }n=1 \end{cases}$ I need to prove that $t(n) \in ...
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1answer
77 views

Proving van Emde Boas recurrence

I have tried to solve the following question: van Emde Boas Bounds Show that $T(u) = T(\sqrt{u}) + O(1)$ has the solution $T(u) = O(\log\log u)$. Hint: consider the binary representation of $u$. ...
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1answer
69 views

Relaxing hypotheses of Master Theorem

This question is related to 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,...
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1answer
61 views

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

Is there a way to solve this recurrence using master theorem: $$T(n) = 4T(n/2) + Θ(n^2/\log n)$$
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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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1answer
614 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 $...
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0answers
41 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, ...
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0answers
178 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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1answer
112 views

Solve recurrence with Master Theorem - Polynomially Smaller/Larger

The problem is to solve the recurrence using Master Theorem : $$T(n) = 2T(n/2)+\log_2 {n}$$ My attempt: $$ a=2, b=2, f(n)= \log_2 {n}, g(n)=n^{\log_b{a}}=n $$ I am torn between case 1 & the ...