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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1answer
26 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
44 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 ...
3
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1answer
59 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
30 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
26 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
47 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
60 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}^{...
1
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1answer
27 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
47 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
46 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
45 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
35 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
64 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
100 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
28 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
34 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 ...
-2
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1answer
39 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
236 views

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

I want to know how to find the complexity of this recurrence.
2
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2answers
77 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? $$ ...
1
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1answer
355 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^...
3
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1answer
58 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 ...
1
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1answer
39 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
154 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 ...
1
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1answer
66 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 ...
2
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1answer
55 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
67 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
93 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
50 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 ...
1
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1answer
57 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 $u$. ...
1
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1answer
51 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,...
2
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1answer
50 views

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

Is there a way to solve this recurrence using master theorem: $$T(n) = 4T(n/2) + Θ(n^2/\log n)$$
2
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0answers
29 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
230 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
30 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, ...
1
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0answers
86 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
79 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 ...
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1answer
38 views

Solving a peculiar recurence relation

Given recurrence: $T(n) = T(n^{\frac{1}{a}}) + 1$ where $a,b = \omega(1)$ and $T(b) = 1$ The way I solved is like this (using change of variables method, as mentioned in CLRS): Let $n = 2^k$ $T(...
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2answers
109 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. ...
2
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1answer
65 views

Time complexity of algorithm inversely proportional to size of sub problem?

Let's say I have an algorithm with time complexity $T_n = T_\frac{n-1}2 + 1$, $T_0 = 0, T_1 = 1$. Assume (Induction hypothesis) $T_n = C\log_2(n+1)$ for some $C$. $T_1$ imposes $C \geq 1$. Therefore ...
3
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1answer
136 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
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1answer
638 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.
1
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1answer
58 views

Why can we ignore the constant factor in Weis's proof of the Master Theorem

In the 4th edition of his Data Structures textbook, Weis gives a proof of part of the Master Theorem. This proof says "Let us ... ignore the constant factor in $\theta(N^k)$ ... I don't understand ...
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0answers
115 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 \\...
2
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0answers
270 views

Why can't we use the Master Theorem on recurrences with floor or ceiling operations? [duplicate]

From my understanding, using such operators on large numbers doesn't have an impact on running time, since the integer rounding becomes negligible after a certain point. For example, the recurrence $$...
2
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1answer
64 views

Can we apply the Master Theorem to the following recurrence?

Our recurrence is $$ T(n)= \begin{cases} T(\lfloor{n/2}\rfloor)+(\log(n))^{2}, & \text{if $n>1$} \\ 1 & \text{if $n=1.$} \end{cases} $$ I have identified $a = 1 > 0$, and $b = 2 > 1$...
1
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2answers
1k views

Recurrence problem T(n) = 2T(n βˆ’ 1) + 1

Can I solve T(n) = 2T(n βˆ’ 1) + 1 using the master theorem method? I don't think it cannot be solved with the master theorem because b=1. Please let me know, if my guess is wrong.
0
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1answer
30 views

How is this equation (involving a recurrence and $\phi(N)$) derived?

As in another question, let $$T(N) = \begin{cases}1 & \text{if } N = 1\\ T(\phi(N)) + \lg(\phi(N))^3 & \text{otherwise} \end{cases}$$ where $\phi(N)$ is Euler's totient function. Tasse ...
0
votes
0answers
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(...
0
votes
0answers
37 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 ...
1
vote
1answer
31 views

Solving recurrence relation where the $f(n)$ has some constant factor $k$ where $0 < k < 1$

I am trying to see if a recurrence relation where $f(n)$ has some constant factor $k$, e.g. $f(n)=kn$ where $0 < k < 1$, is $O(n)$. I am reaching a different result depending which route I take. ...