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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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Recurrence Upper Bound Estimation

I'm going through CLRS and was trying to solve for the asymptotic bound of the following recurrence (exercise 4-5.4) $$T(n) = 4T(n/2) + n^2\text{lg }n$$ According to CLRS definition of Master Theorem, ...
Jackson Schuetzle's user avatar
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1 answer
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Big O notation of T(n) = T(n/2) + O(log n) using master theorem?

I am aware that the algorithm has 1 recursive call of size n/2 and the non-recursive part takes O(log n) time. Master theorem formula is T(n) = aT(n/b) + O(n^d). In this case a = 1, b = 2, but I am ...
inkwad's user avatar
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Solutions to Recurences

I am currently learning various techniques in order to solve recurrences. One of which is the generalized master's theorem. The current problem I am attempting is as follows $H(n) < 4H(2n/5) + H(...
kusakus's user avatar
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$F(n, L) = 2F(n / 2, L) + nL + n^2 log(n)$ and Master Theorem

I have the following recurrence: $F(n, L) = 2F(n / 2, L) + nL + n^2 log(n)$. Am I correct in saying that $F(n, L) \in O(n \log(n) L + n^2 log(n))$? I got to this result by bounding the $nL$ and $n^2 \...
Jovan Komatovic's user avatar
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22 views

Is this a special case of a recurrence where the Master Method is not applicable?

So in an exam, this was the recurrence: $$ T(n) = 2T(n/2) + n log(n) -n + O(log(n))$$ $$T(1) = 1$$ Why does the master method not apply here? I think it is indeed int he form $$aT(n/b) + f(n)$$ You ...
Yuirike's user avatar
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106 views

Complexity of recursive function using Master theorem

this code aims to determine whether there exists a contiguous subarray starting from index 0 in the given array A whose elements sum up to the target value S. can we apply Master theorem to find out ...
Arugo's user avatar
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Evaluating $T(n) = 4T(\frac{n}{5}) + \log n$: Master Theorem vs. Recursion Tree

I'm wondering where (how? why?) my reasoning (by imagining the recursion tree) deviates from the application of the Master Theorem (Case 1) to this recurrence. The Master Theorem gives $\Theta(n^{\...
Per48edjes's user avatar
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1 answer
38 views

What is the "big theta" order of the solution of T_n = T_(n/2) + log n, n > 0?

What method(s) could be used to solve this? I am still new to this stuff and would appreciate detailed justification for every step as well as some intuition and the examination of all possible viable ...
user79644's user avatar
3 votes
1 answer
374 views

Time complexity of tree algorithm

I'm new to recurrence relations and master theorem so trying to learn. Say there's an algorithm $A$ whose input is the root of a binary tree $T$. $A$ recurses so that it's called on each and every ...
onepiece's user avatar
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Does the Master Theorem apply to T(n) = 3T(n/3) + n/log2(n)?

Id say this is the first case of Master Theorem, but when I try to prove that the limit of f(n)/ n ^ (1-E) is 0, I cannot do it. Does anyone have a solution?
Mara F's user avatar
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Can this reccurrence recurrence be solved using Master Theorem?

Assume we have: $$T(n)=7T(\frac{n}{2})+n^2\lg{n}$$ Can we solve it using master theorem? As we know $n^{\lg_2{7}}\approx n^{2.81}$. On the other hand, we have $f(n)=n^2\lg n$. So we should compare $n^....
Ferran Gonzalez's user avatar
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1 answer
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find $f(n)$ for recurrence $T(n)=2T(\dfrac{n}{2})+\mathcal{O}(n\log{n})=\Theta(f(n))$

We have recurrence $T(n)=2T(\dfrac{n}{2})+\mathcal{O}(n\log{n})$ and assume $T(1)$ is a constant. Find asymptotically tight bounds $\Theta(f(n))$ for $T(n)$. There's something that confuses me. We ...
Mason Rashford's user avatar
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2 answers
187 views

Prove $T(n)=10T(\frac{n}{3})+n\sqrt{n}=\Theta(n^{\lg_3{10}})$ using induction

We have this recurrence: $$T(n)=10T(\frac{n}{3})+n\sqrt{n}.$$ We can solve it using Master Theorem and say it is $\Theta(n^{\log_3{10}})$. I want to prove it using induction but I don't know the ...
Mason Rashford's user avatar
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32 views

Master Theorem - Solving Recurrence

I've been stuck for hours trying to solve the recurrence $T(n) = 7T(n/3) + n^2 + 2n$ by using case 3 of the master theorem. I've done a good chunk of the proof, but currently stuck attempting to solve ...
Jeremy Bowens's user avatar
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93 views

The master theorem soution to T(n) : T(n/4) + logn

When i tried to find the time complexity of this recurrence relation with the master theorem, I got log^2n, but I'm told that it's logn. I used the masters theorem, for this case.. a=b^k (1=4^0) ... ...
Yor's user avatar
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4 votes
2 answers
1k views

Does T(n) = 2 · T(2n) + n apply to Master method?

I'm trying to apply the master method to the following recurrence: $$T(n) = 2 \cdot T(2n)+n.$$ We have $a=2$ and $b=1/2$. Also, $f(n)=n$ and $n^{\log_b a} = n^{\log_{1/2} 2} = n^{-1}$ since $\log_{1/2}...
Jarvis's user avatar
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Solving a recurrence relation using the Master Theorem

I'm trying to solve this recurrence relation: $T(n) = T(\frac{n}{2}) + T(\frac{n}{5}) + T(\frac{n}{10}) + c_1n$ ; n > 1 $T(n) = c_2n$ ; n = 1 My first thought was to combine the fractions and ...
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1 answer
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Why not $O(n^{\log_ba})$ for case 1 of the Master Theorem instead of $O(n^{(\log_ba) - \epsilon})$?

Someone who was explaining to me the master theorem said that for the case 1, we compare the $n^{\log_b(a)}$ and $f(n)$. If the growth rate of $n^{\log_b(a)}$ is greater than the growth rate of $f(n)$ ...
Mina's user avatar
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In the Master Theorem, if one term is smaller than another, can we drop it from the equation and use big O instead of theta?

Considering the runtime analysis (with the master theorem) of the function below $T(n) = 12T(\frac{n}{4}) + 2\sqrt{n} + \log^4(n)$. As I could not figure out a way to get the equation in the form $T(...
Mini's user avatar
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How to solve T(n)=2T(√n)+(loglogn)^2?

Trying to solve the recurrence, but no clue how to deal with the (loglogn)^2 part
Chris W's user avatar
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1 answer
113 views

Solve Recurrence T(n) = 4T(n/4) + n*[log(n)]^2

I am trying to solve T(n) = 4*T(n/4) + n*[log(n)]^2 I decided to use Master Theorem so I found a,b=4 and logb(a)=1. I thought that 3rd case is the solution but I ...
Grigorios Garoufalis's user avatar
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1 answer
572 views

Find matrix local minimum - two analysis which seem to get contradictory runtimes

Suppose you have an $n\times n$ matrix and you want to find a local minimum. To find it you scan the middle row and column and identify a minimum. If it is a local minimum, you're done; if not, you ...
Addem's user avatar
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3 votes
2 answers
601 views

Asymptotic Analysis of T(n) = 2T(n/8) + 2T(n/4) + n

Given the recurrence $$T(n) = 2T\bigg(\frac{n}{8}\bigg) + 2T\bigg(\frac{n}{4}\bigg) + n$$ My professor says that $T(n)$ is $O(n\log n)$ but I have calculated a complexity of $O(n)$ as shown below with ...
Bender's user avatar
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Solve the recurrence $3T(n) = T(n/3)+ \sqrt{\log n}$

How can you solve the recurrence $$3T(n) = T(n/3)+ \sqrt{\log n}$$ using the master theorem? I am lost in this question.
tjbeast's user avatar
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121 views

Big theta and big 0 bounds for iteration method and Master Theorem

In Algorithms 1, I'm noticing that big-Theta running times are always used for recurrence relations when using the iteration method. Meanwhile, using the Master Theorem always seems to result in a big-...
There's user avatar
  • 111
1 vote
2 answers
152 views

Solve the recurrence equation $T\left(n\right)=\sqrt{n}\cdot T\left(\sqrt{n}\right)+c\log n$

I tried to solve the recurrence $T\left(n\right)=\sqrt{n}\cdot T\left(\sqrt{n}\right)+c\log n$ using the Master Theorem. I tried the following way: $n = 2^k$ $2^{\frac{2}{k}}\cdot T\left(2^k\right)+\...
LoveYourz's user avatar
0 votes
2 answers
568 views

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?
Vincenzo Iannucci's user avatar
1 vote
2 answers
117 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 ...
20210352's user avatar
2 votes
3 answers
207 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 ...
matti1499's user avatar
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1 answer
592 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 $T(n) = O(n \log n)$?
Vincenzo Iannucci's user avatar
1 vote
1 answer
154 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 ...
Jack's user avatar
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1 answer
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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 ...
five_star_021's user avatar
1 vote
1 answer
156 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 ...
Parviz Pirizade's user avatar
0 votes
2 answers
49 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 ...
Vil's user avatar
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0 votes
0 answers
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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 ...
super.t's user avatar
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2 votes
3 answers
200 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? $$T(n)= 3 \cdot T(\sqrt[3]{n})+n^3$$
Preguntador's user avatar
1 vote
3 answers
981 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? $T(n)=4T(\sqrt{n}/3)+(\log n)^2$
maryam ghanbari's user avatar
1 vote
2 answers
227 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 ...
lambduh's user avatar
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-1 votes
2 answers
124 views

Divide and conquer recurrence relation

I have divide and conquer problem and below is the recurrence relation for it $$\begin{align}t (n) &= a\cdot t (n/4) + O (n^2/\log(n)) + O(n^2)\\ t(n) &= a\cdot t (n/4) + O(n^2) \end{align}$$ ...
Zero One's user avatar
1 vote
1 answer
47 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 ?
Vighnesh Kumar's user avatar
0 votes
2 answers
61 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 ...
Diana 's user avatar
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1 vote
3 answers
56 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 ...
Diana 's user avatar
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3 votes
1 answer
209 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)^{\...
random0620's user avatar
2 votes
1 answer
272 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 $...
TheCollegeStudent's user avatar
3 votes
1 answer
7k 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 $f(n)=\Theta(n^{\log_ba}\log^kn)$, ...
Dan's user avatar
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0 votes
1 answer
65 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 ...
kasra's user avatar
  • 235
1 vote
1 answer
1k 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}^{...
Vincenzo Iannucci's user avatar
1 vote
1 answer
176 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.
mmafshari's user avatar
1 vote
1 answer
4k 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, ...
Vincenzo Iannucci's user avatar
2 votes
1 answer
168 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 ...
heretoinfinity's user avatar