Questions tagged [recurrence-relation]
a definition of a sequence where later elements are expressed as a function of earlier elements.
483
questions
2
votes
1answer
39 views
Solving a recurrence relation involving square roots
Give an asymptotic upper bound for $$T(n) = \sqrt{n}·T(\sqrt{n})+n+n/\log n. $$
How can I solve this recurrence relation, which involves square roots?
-1
votes
1answer
53 views
Recurrence Problem T(n) = 3T(n/3) + n
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:
$$T(n) = 3T(\frac{n}{3}) + n$$
How can I ...
0
votes
1answer
25 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 ...
1
vote
1answer
73 views
Solving the recurrence relation T(n) = 2T(n/2) + nlog n via summation
I have seen a few examples of using the master theorem on this to obtain O(n*log^2(n)) as an answer. I am trying to solve this by unrolling and solving the summation, but I can't seem to get the same ...
0
votes
1answer
30 views
Please provide me a solution of Max-Heapify using Recursion Tree
I tried my best to solve the recurrence relation.
$T(n) \le T(2n/3) + \Theta(1)$
Using the recursion tree.
I could reach out the boundary condition when at depth ...
0
votes
1answer
30 views
Solving recurrences (tree method) with square roots
I am trying to find the upper and lower bounds for this recurrence, but I am not sure how to handle to square root:
$$
T(n) = 4T(n/2) + n^2\sqrt{n}
$$
3
votes
1answer
67 views
Solve recurrence relation that depends on depth of recursion
The specific problem I'm working on is the puzzle presented in this video. For those who don't want to watch the video, my summary of the puzzle is:
A frog is sitting on the edge of a pond facing ...
0
votes
0answers
23 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(...
1
vote
1answer
20 views
Complexity of iterative exponentiation
I've watched multiple videos and read articles about recursion but I'm still confused. I've got this problem here but I'm unsure how to answer it:
The following function calculates $x^n$ ...
0
votes
1answer
22 views
Recurrent relation for algorithms with two stages
I am trying to do the recurrence relation for my algorithm, but it has two variables $T(n,m)$. For sufficiently small $n$, $m$ is practically the same as $n$, but $m$ cannot grow beyond some constant $...
1
vote
1answer
23 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. ...
2
votes
1answer
31 views
evaluating time complexity of a code
I'm trying to evaluate the time complexity of the following :
...
1
vote
0answers
21 views
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 ...
1
vote
2answers
36 views
Complexity guess and induction proof
I was trying to prove by induction that
$$
T(n) = \begin{cases}
1 &\quad\text{if } n\leq 1\\
T\left(\lfloor\frac{n}{2}\rfloor\right) + n &\quad\text{if } n\gt1 \\
\end{...
1
vote
1answer
22 views
Recurrence Equation upper limit problem
I was looking at my teacher's notes and came about the following recurrence equation :
$$
T(n) =
\begin{cases}
1 &\quad\text{if } n\leq 1\\
4T\left(\frac{n}{2}\right) + n^3 ...
0
votes
2answers
29 views
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 ...
1
vote
1answer
47 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....
2
votes
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\...
1
vote
1answer
93 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-...
2
votes
1answer
349 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\...
1
vote
1answer
110 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:...
2
votes
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.
2
votes
2answers
50 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 ...
0
votes
1answer
32 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.
2
votes
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:
$$
T_n = 8^{\left(...
2
votes
1answer
58 views
How to prove a bound function for a sequence of numbers?
Let $G_n$ be defined by
$$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}$$
How can I prove that $f(n) = n$...
1
vote
1answer
40 views
Solve the recurrence $a_n - 3a_{n-1} + 2a_{n-2} = 6 \cdot 2^n$
Consider the recurrence
$$ 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 \\
...
0
votes
0answers
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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
55 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?
$$T(n) = \begin{cases}
1 & n = 1\\
5T(n-1) + n^2 & \text{otherwise}
\end{cases}$...
2
votes
2answers
181 views
Finding recurrence relation for running time of an algorithm
I am pretty new to this, consider the following algorithm:
...
0
votes
0answers
33 views
Substitution method for $T(n) = 2T(7n/10) + O (1)$ [duplicate]
I want to solve $T(n) = 2T(7n/10) + O (1)$ using the substitution method.
I think the solution should be $T(n) = O(n\log n)$, but I am having trouble constructing a proof by substitution.
3
votes
2answers
94 views
How can I solve $T(n) = 2T(\sqrt{n-1} + 2) + 1$ recurrence using tree method?
The recurrence I have is
$T(n) = 2T(\sqrt{n-1} + 2) + 1$
I don't know how to solve it. I didn't find much on the internet with square roots in recurrences especially with constants inside of it. I'm ...
1
vote
0answers
83 views
What are the asymptotic bounds (upper bound on time complexity) of the following function?
I am trying to find the upper bound on time complexity of the recursive function defined by the following equation:
$$Q(t) = \sum^{N}_{i=1} q_i \big(g_i^{\frac{1}{m-1}} + Q(t+1)^{\frac{m}{m-1}}\big)^{\...
0
votes
2answers
59 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 $T(n)$...
3
votes
0answers
47 views
Worst Case Analysis of a Multivariate Recurrence of a Graph Algorithm
I have a graph algorithm that runs in:
$$ T(n, m) = \begin{cases}
c_1 & n \leq 2 \lor m = 1\\
T(n - i,\ m - j - k) + T(i, k) + c_2 m + c_3 n & m \leq (n-i)i\\
T(n - i,\ m) + T(i, m) + ...
1
vote
3answers
90 views
Solving $T(n)=4T(n/2)-1$ without using the master theorem [duplicate]
How can I solve the following recurrence without using the master theorem?
$T(n)= 4T(n/2)-1$ for $n>4$ and $T(n)=5$ for $n\le 4$, $n$ is a power of $2$.
1
vote
3answers
117 views
Solving recurrence equation $T(n)=T(n^{2/3})+17$
How can the following recurrence equation be solved by one of three main ways:
$$T(n)=T(n^{2/3})+17$$
I have tried solving it by the iteration way. However it does not work for me since I can't find ...
0
votes
2answers
177 views
How to find the big o running time if the recursion function have different cases of recursion with different fraction of n?
How to find the big o running time if the recursion function have different cases of recursion with different fraction of n?
If I have a recursive function like this for example (This is just an ...
1
vote
4answers
332 views
Deriving lower and upper bounds for T(n) = T(n-1) + T(n-2) + 10
The solution is to find the upper and lower bounds from:
2T(n-2) < T(n) < 2T(n-1) + 10
So I have to find
...
2
votes
1answer
57 views
Generalizing Knuth's $O(\log_2 n)$ Fibonacci algorithm to linear homogenous recurrences
Knuth has a neat algorithm that uses matrix exponentiation to compute the $n$th Fibonacci number in $O(\log_2 n)$-time 1. However, there doesn't seem to be a lot of resources on generalizing his idea ...
0
votes
0answers
16 views
Solve Recurrence $T(n) = T(pn) + T((1-p)n) + \Theta(n)$ [duplicate]
For $0 < p < 1$, how can you solve the recurrence $$T(n) = T(pn) + T((1-p)n) + \Theta(n)$$ using the substitution method. My guess is $T(n) = O(n \log n)$, but plugging this guess in leads to a ...
1
vote
1answer
42 views
Solving the recurrence $T(n) = n^{3/4}𝑇(𝑛^{1/4})+ n $
I need to solve the following recurrence relation: $T(n) = n^{3/4}𝑇(𝑛^{1/4})+ n $. Obviously, the master theorem doesn't apply here so I was using the substitution method. I used $x=\log n$ and $F(x)...
7
votes
2answers
119 views
The recursion $T(n) = T(n/2)+T(n/3)+n$
I'm looking at the reccurrence
$$T(n) = T(n/2) + T(n/3) + n,$$
which describes the running time of some unspecified algorithm (base cases are not supplied).
Using induction, I found that $T(n) = O(n\...
1
vote
0answers
33 views
Prove that the upper bound for T(n)=T(an)+T(bn)+O(n) is O(n) [duplicate]
While learning Median of Medians algorithm i came across the following lemma ;
"For any recurrence of the form $T(n)<=T(an)+T(bn)+O(n) $,
if $(a+b)<1$ the reccurence will solve to $O(n)$"
(...
2
votes
1answer
39 views
Solving T(n) = T(n-1)*T(n-2)
So, this is how I solved
$\displaystyle
T(n-1) \approx{} T(n-2)
$
$\displaystyle
T(n) = T(n-1)^2
$
Add log in both sides
$\displaystyle
log(T(n)) = 2log(T(n-1))
...
1
vote
1answer
58 views
Master Method: $T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$
I'm having a hard time trying to understand how to solve this recurrence relation using the Master Method:
$$T(n) = 10T\Big(\frac{n}{2}\Big) + \frac{n^4}{\log(n)}$$
First, we have:
$a = 10,\ b = 2$ ...
1
vote
1answer
105 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)?
2
votes
2answers
361 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, ...
2
votes
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.
...
