Questions tagged [recurrence-relation]

a definition of a sequence where later elements are expressed as a function of earlier elements.

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Useful conditions for proving super polynomial lower bound for some kind of recurrences

Given a recurrence of the form $\forall n,m.\ \ T(n,m)=\begin{cases}1,&,m=1\\\sum_i{T(n_i,m_i)}&,\text{else}\end{cases}$ Note: both $n_i$ and $m_i$ are dependent on $n,m$ so they should have ...
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32 views

Complexity guess and induction proof

I was trying to prove by induction that $$ T(n) = \begin{cases} 1 &\quad\text{if } n\leq 1\\ T\left(\lfloor\frac{n}{2}\rfloor\right) + n &\quad\text{if } n\gt1 \\ \end{...
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1answer
21 views

Recurrence Equation upper limit problem

I was looking at my teacher's notes and came about the following recurrence equation : $$ T(n) = \begin{cases} 1 &\quad\text{if } n\leq 1\\ 4T\left(\frac{n}{2}\right) + n^3 ...
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Converting a Recurrence Relation to its Closed Form [duplicate]

I have a recurrence relation of the form given below (taken from Analysis of Algorithms - An Active Learning Approach by Jeffrey J. McConnell): $T(n) = 2T(n - 2) - 15 $ $T(2) = T(1) = 40 $ I am ...
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1answer
21 views

Akra-Bazzi method integral diverges

I want to solve this recursion: $$T(n) = 5T(\frac{n}{5}) + \frac{n}{lg(n)}$$ My attempt and issue: None of the cases for master theorem apply here. I tried using Akra-Bazzi method (https://en....
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2answers
39 views

How do we guess the recurrence relation from the given equation

In this book introduction to algorithms , i have been reading about a method named substitution method to solve the recurrence, the recurrence equation is \begin{equation} T(n)=2 T(\lfloor n / 2\...
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1answer
60 views

Upper bound $T(n) = 9T(\sqrt[3]{n}) + O(1)$

The problem is this: Use the recursion-tree method to give a good asymptotic upper bound on $$ T(n) = 9T(\sqrt[3]n) + \Theta(1). $$ I am able to get the tree started and find a pattern with the sub-...
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1answer
342 views

Solving recurrence relation with minimum and factorial

I need to solve the following recurrence relation, where $T(n,m)$ is defined over $\Bbb N_+\times\Bbb N_+$. $T(n,m)=\begin{cases} 1, & n=1\text{ or }m\leq 2(n-1)!\\ \min\limits_{a,b,c\geq 1,\ c\...
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1answer
106 views

Recurrence with Minimum

I need to solve the following recurrece: $T(n,m)=\begin{cases} 1, & m\leq 2(n-1)!\\ \min\limits_{a,b\geq 1\\a\cdot b\leq (n-1)!}{T(n-1,a)+T(n-1,b)+T(n,m-ab)}, & \text{else} \end{cases}$ Note:...
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1answer
54 views

solving the recurrence t(n)=t(n-2)+d*(n^2)/2 with iteration method

How can I solve $$T(n)=T(n-2)+\frac {d}{2}n^2$$ I couldnt find $d$ (dont know if I have to) and after 3 iterations I got to $k= \frac{n-1}{2}$ but had trouble to continue.
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46 views

Number of possible heaps on $\{1,…,2^h-1\}$

Let $C_h$ be the number of possible heaps for the set of keys $\{1,...,2^h-1\}$. Determine a recurrence relation for $C_h$ via the substitution method and prove it. Definition A binary tree ...
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1answer
31 views

Trouble finding what this recurrence solves to [duplicate]

I have a recurrence relation of the form $T(n) = 2T(n/2)+O(1)$ I'm not sure how to deal with the big $O$-notation in the problem in order to start solving it ? Any help would be appreciated.
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1answer
55 views

What is the closed-form expression for $T_n = \left(\sum_{i=1}^{n-1}7 T_i\right) + 1$ where $T_1 = 1 ?$ [closed]

Problem: Find the closed-form expression for$$ T_n = \left(\sum_{i=1}^{n-1}7 T_i\right) + 1 \tag{1} $$where $T_1 = 1 .$ Calculating this sum I came up with the following result: $$ T_n = 8^{\left(...
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1answer
53 views

How to prove a bound function for a sequence of numbers?

Let $G_n$ be defined by $$G_n = \begin{cases} 1 & n=0 \\ 2 & n = 1 \\ 3 & n = 2 \\ 4 & n = 3 \\ 2G_{n-1}-2G_{n-3}+G_{n-4} & n\geq4 \end{cases}$$ How can I prove that $f(n) = n$...
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39 views

Solve the recurrence $a_n - 3a_{n-1} + 2a_{n-2} = 6 \cdot 2^n$

Consider the recurrence $$ a_n - 3a_{n-1} + 2a_{n-2} = 6 \cdot 2^n. $$ I tried to solve this as follows. First, I found the homogeneous solution: $$ a_n^{(h)} = r^2 - 3r + 2r \\ (r-2)(r-1) = 0 \\ ...
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36 views

How can I prove the linear time search algorithm takes O(n) time? [duplicate]

The recurrence relation for the algorithm is an eccentric form that has an additional term: $T(n) = T[\frac{n}{2}] + T[\frac{7n}{10} + 6] + n$. Exactly how can I prove that this recurrence relation ...
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23 views

Meaning of polynomially larger, / smaller and meaning of polynomial larger / polynomial smaller

Hi I have spent the last two hours trying to find a general definition for both of these terms, but it seems like in computer science, people do not really agree on definitions. And I haven't even ...
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1answer
52 views

Closed form solution of $T(n) = 5T(n-1) + n^2$ [duplicate]

How to find the closed form solution of this equation? Is there a repeatable pattern for solving this equation? $$T(n) = \begin{cases} 1 & n = 1\\ 5T(n-1) + n^2 & \text{otherwise} \end{cases}$...
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2answers
85 views

Finding recurrence relation for running time of an algorithm

I am pretty new to this, consider the following algorithm: ...
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28 views

Substitution method for $T(n) = 2T(7n/10) + O (1)$ [duplicate]

I want to solve $T(n) = 2T(7n/10) + O (1)$ using the substitution method. I think the solution should be $T(n) = O(n\log n)$, but I am having trouble constructing a proof by substitution.
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How can I solve $T(n) = 2T(\sqrt{n-1} + 2) + 1$ recurrence using tree method?

The recurrence I have is $T(n) = 2T(\sqrt{n-1} + 2) + 1$ I don't know how to solve it. I didn't find much on the internet with square roots in recurrences especially with constants inside of it. I'm ...
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81 views

What are the asymptotic bounds (upper bound on time complexity) of the following function?

I am trying to find the upper bound on time complexity of the recursive function defined by the following equation: $$Q(t) = \sum^{N}_{i=1} q_i \big(g_i^{\frac{1}{m-1}} + Q(t+1)^{\frac{m}{m-1}}\big)^{\...
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57 views

Can you always prove the asymptotic bound of a recurrence of the form aT(n/b) + f(n) using the substitution method?

To make my question more concrete, here is an example I am stuck on. I want to prove that $T(n) = 8T(\frac{n}{2}) + n^3$ is asymptotic bound by $n^3\log(n)$ using the substution method. That is $T(n)$...
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Worst Case Analysis of a Multivariate Recurrence of a Graph Algorithm

I have a graph algorithm that runs in: $$ T(n, m) = \begin{cases} c_1 & n \leq 2 \lor m = 1\\ T(n - i,\ m - j - k) + T(i, k) + c_2 m + c_3 n & m \leq (n-i)i\\ T(n - i,\ m) + T(i, m) + ...
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3answers
77 views

Solving $T(n)=4T(n/2)-1$ without using the master theorem [duplicate]

How can I solve the following recurrence without using the master theorem? $T(n)= 4T(n/2)-1$ for $n>4$ and $T(n)=5$ for $n\le 4$, $n$ is a power of $2$.
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110 views

Solving recurrence equation $T(n)=T(n^{2/3})+17$

How can the following recurrence equation be solved by one of three main ways: $$T(n)=T(n^{2/3})+17$$ I have tried solving it by the iteration way. However it does not work for me since I can't find ...
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2answers
72 views

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n?

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n? If I have a recursive function like this for example (This is just an ...
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4answers
169 views

Deriving lower and upper bounds for T(n) = T(n-1) + T(n-2) + 10

The solution is to find the upper and lower bounds from: 2T(n-2) < T(n) < 2T(n-1) + 10 So I have to find ...
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1answer
56 views

Generalizing Knuth's $O(\log_2 n)$ Fibonacci algorithm to linear homogenous recurrences

Knuth has a neat algorithm that uses matrix exponentiation to compute the $n$th Fibonacci number in $O(\log_2 n)$-time 1. However, there doesn't seem to be a lot of resources on generalizing his idea ...
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Solve Recurrence $T(n) = T(pn) + T((1-p)n) + \Theta(n)$ [duplicate]

For $0 < p < 1$, how can you solve the recurrence $$T(n) = T(pn) + T((1-p)n) + \Theta(n)$$ using the substitution method. My guess is $T(n) = O(n \log n)$, but plugging this guess in leads to a ...
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1answer
31 views

Solving the recurrence $T(n) = n^{3/4}𝑇(𝑛^{1/4})+ n $

I need to solve the following recurrence relation: $T(n) = n^{3/4}𝑇(𝑛^{1/4})+ n $. Obviously, the master theorem doesn't apply here so I was using the substitution method. I used $x=\log n$ and $F(x)...
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83 views

The recursion $T(n) = T(n/2)+T(n/3)+n$

I'm looking at the reccurrence $$T(n) = T(n/2) + T(n/3) + n,$$ which describes the running time of some unspecified algorithm (base cases are not supplied). Using induction, I found that $T(n) = O(n\...
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Prove that the upper bound for T(n)=T(an)+T(bn)+O(n) is O(n) [duplicate]

While learning Median of Medians algorithm i came across the following lemma ; "For any recurrence of the form $T(n)<=T(an)+T(bn)+O(n) $, if $(a+b)<1$ the reccurence will solve to $O(n)$" (...
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1answer
38 views

Solving T(n) = T(n-1)*T(n-2)

So, this is how I solved $\displaystyle T(n-1) \approx{} T(n-2) $ $\displaystyle T(n) = T(n-1)^2 $ Add log in both sides $\displaystyle log(T(n)) = 2log(T(n-1)) ...
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1answer
51 views

Master Method: $T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$

I'm having a hard time trying to understand how to solve this recurrence relation using the Master Method: $$T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$$ First, we have: $a = 10,\ b = 2$ ...
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1answer
72 views

Solve the recurrence relation T(n)=3T(√n)+lg(n) [duplicate]

Master's Theorem is known to me, but I can't understand how to apply this theorem to this problem. So, how I will find Θ of T(n)?
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2answers
263 views

Recurrence relation of quicksort depending on its pivot

I understand how the recurrence relation of quicksort is $T(n) = 2T(n/2)+\mathcal{O}(n)$, but if we are guaranteed a certain pivot, for example $n/4$th smallest element to be the pivot every time, ...
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1answer
32 views

Maximum Expected Fishing Day (Recurrence Relation)

John joined a meetup where organize day long fishing trip once a month. The organizers are vary poor at planning, so will organize fishing on a random day of the month without any advance notice. ...
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1answer
36 views

Using master theorem to solve recurrence with log [duplicate]

I'm not sure how to solve apply the master theorem in order to solve this recurrence: $$ T(n) = 4T(n/3) +O(n\log n),\text{ where } T(1) = 1.$$ The master theorem I have been shown is normally ...
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1answer
28 views

Find the computational complexity of the given program

int seq(int n) { if(n == 0 || n == 1) return n; return(seq(floor(n/2)) + seq(ceil(n/2)); } Find the computational complexity of the above program. ...
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1answer
46 views

Computing number of ways to make change

Given a list $C=[c_1,c_2,\dots,c_k]$ of positive integers, representing the values of $k$ varieties of coins, and a positive integer $n$, let $f(n,C)$ be the number of handfuls of coins with total ...
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1answer
79 views

How to write recurrence relation for backtracking problem?

I am not able to understand how to write a recurrence relation for n queen problem. I searched on web and everywhere it was given directly without explaining how can we arrive to that. Recurrence ...
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40 views

How do I show that an iterative solution to Tower of Hanoi performs the same exact steps as a recursive solution? [duplicate]

So given the typical recursive solution to the Tower of Hanoi problem wherein you reduce the n-disk tower to two instances of an (n-1)-disk tower i.e move (n-1) disks from start to auxiliary. move ...
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1answer
124 views

Induction to prove equivalence of a recursive and iterative algorithm for Towers of Hanoi

Using induction how do you prove that two algorithm implementations, one recursive and the other iterative, of the Towers of Hanoi perform identical move operations? The implementations are as follows....
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2answers
409 views

Solving $T(n) = 3T(n-1) + 2$

I am trying to get better at solving recurrence relations, so I am making my own simple relations and try to solve them. I have made the following recurrence: $$T(n) = 3T(n-1) + 2, \quad\quad T(1) = ...
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1answer
81 views

Solving recurrence relations with two variables

I am trying to solve this recurrence relation with two variables: $$T(n, k) = T(n - 1, k - 1) + T(n - 1, k)$$ The base cases are: $T(n, k) = 1$ if $k = 0$ $T(n, k) = 0$ if $k > n$ I was ...
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3answers
437 views

Solve recurrence T(n)=2T(n-1)+n for n greater than 1 and T(1)=1 [duplicate]

Problem statement: Solve $T(n)$ for $T(n)=2T(n-1)+n$, $n > 1$, and $T(1)=1$. My attempt: I tried back substituting but I am unable to find a general pattern: $$\begin{align*} T(n) &=2^2 T(n-2)...
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1answer
76 views

Proof of a lower bound of the recurrence relation (the CLRS's 4.6-2 exercise)

I am trying to find a solution to the ex. 4.6-2 of the Introduction to Algorithms by Cormen, Leiserson, Rivest, Stein (the third edition). It requires, for recurrence relations $T(n)=aT(n/b)+f(n)$ ...
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0answers
30 views

Recurrence Relation for Column Major Form of multidimensional array

A two dimensional array is stored in column major form in memory if the elements are stored in the following sequence $$A[0][0] A[1][0] A[2][0]...A[n_1-1][0] ... A[0][1] A[1][1] ... A[n_1-1][1] .... A[...
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1answer
51 views

Can I say the two cases of Recursion Tree are always either $\theta{(n)}$ or $\theta({n\log{n}})$

Given positive constants: $c_1, c_2, ..., c_k, c^\prime$, assume that $T(n) = T(c_1n) + T(c_2n) + ...+ T(c_kn) + c^\prime n$ There are two cases: $c_1 + c_2 + ...+ c_k < 1$ $c_1 + c_2 + ...+ c_k ...