0
votes
1answer
23 views

Analyze the following recurrence relation.

I'd like to express the following recurrence relation as the number of multiplications it performs, and then again but this time as the number of additions/subtractions it performs. ...
1
vote
1answer
73 views

Solving recurrence relation: f(n) = 3f(n/2) - 2f(n/4) | f(2) = 5, f(1) = 3

f(n) = 3f(n/2) - 2f(n/4) | f(2) = 5, f(1) = 3 I have attempted to solve it by letting n = 2k f(2k) = 3f(2k-1) - 2f(2k-2) Then set S(k) = f(2k) S(k) = 3*S(k-1) - 2*S(k-2) ...
0
votes
1answer
31 views

Why does $T(n) \leq 2 T(\lceil \frac{n}{2} \rceil)+\mathcal{O}(n)$ imply $T(n)=\mathcal{O} (n\log(n))$

I am learning about sort algorithms and their complexities. For merge sort, I'm confronted with $T(n) \leq 2 T(\lceil \frac{n}{2} \rceil)+\mathcal{O}(n)$. The author makes the claim with no ...
1
vote
4answers
46 views

Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...
2
votes
4answers
116 views

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+…+T(\frac n {2^k})$

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+...+T(\frac n {2^k})$ while k is some constant and for any $n\leq3$ $\ T(n)=c$ for k=1 ...
0
votes
1answer
28 views

Big-O evaluation:

I have the expression: $$f_{k}(n,m) = (n - k)(m - k) + f_{k+1}(n,m)$$ which runs until k = n or m. What is the big theta of this function in terms of n,m? A naive approach is to assume that m does ...
3
votes
1answer
2k views

Upper bound for $T(n) = T(n - 1) + T(n/2) + n$ with recursion-tree

I'm reading through Introduction to Algorithms, 3rd ed. and I got stuck on the following recurrence (exercise 4.4-5): $$T(n) = T(n - 1) + T(n/2) + n$$ The exercise asks you to find the upper bound ...
0
votes
1answer
160 views

Solving a recurrence relation with floors and comparing it with other complexity classes

The problem that I am struggling with is the recurrance relation $T(n) = \lfloor(T(n/2))\rfloor + \lfloor(log \space n)\rfloor$ Where $T(1) = 1$ I am supposed to answer true/false to each of the ...
2
votes
0answers
68 views

Solving recurrence relation of algorithm complexity?

Supposing I write an algorithm that results into this kind of recurrence relation $$\left\{ \begin{array}{ll} T(0)=T(1)=1 \\ T(n)=T\left(\lfloor n/2 \rfloor \right)+T\left(\lceil n/2 ...
0
votes
1answer
298 views

Solving a recurrence relation using Z transform

I'm trying to solve the following recurrence using Z transforms: For $n\in \mathbb{N}^{*}$ $T(n)=1\ for\ n< 4$ $T(n)=T(\lfloor \frac{n}{4} \rfloor)+T(\lfloor \frac{3n}{4} \rfloor)+n\ for\ n\geq ...
0
votes
1answer
147 views

Solving the following recurrence relation

I have a recurrence relation, it is like the following: $$ T(e^n) = 2\cdot T(e^{n-1}) + e^n, \text{ where $e$ is the natural logarithm} $$ To solve this and find a Θ bound, i tried the following: I ...
1
vote
2answers
90 views

The use of master theorem appriopriately

I have a recurrence relation and trying to use master theorem to solve it. The recurrene relation is: $$T(n) = 3T\left(\tfrac n5\right) + \sqrt n$$ Can i use the master theorem in that relation? If ...
1
vote
2answers
742 views

Solving a recurrence relation using master method

I know that the Master theorem is used for the recurrence relations of the form: $$T(n) = aT(n/b) + f(n)$$ In my question, I am supposed to solve the following recurrence relation by using Master ...
1
vote
2answers
170 views

Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n)=4T\left(\frac{n}{2}\right)+n^2$$ My guess is $T(n)$ is $\Theta (n\log n)$ (and I am sure about it ...
3
votes
1answer
155 views

Issue while applying Master Theorem

I've read about the master theorem for solving recurrences in Introduction to Algorithms, but have a problem (probably, due to misunderstanding) while applying it in some cases. For example, having ...
5
votes
0answers
512 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
2
votes
0answers
66 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
votes
2answers
349 views

Solving recurrences of the form $T(n) = aT(n/a) + \Theta(n \log_2 a)$

The time complexity for the merge sort algorithm is $T(n) = 2T(n/2)+\Theta(n)$, and the solution to this recurrence is $\Theta(n\lg n)$. However; assume you are not dividing the array in half for ...
2
votes
4answers
217 views

Analysis of Algorithms: Solving Recursion equations: $\quad T(n)= T(cn)+T(dn)+n$

How can I prove that the solution for the following recursion equation is $\Theta(n)$: $$T(n)= T(cn)+T(dn)+n \text{ for } d,c>0 \text{ and } c+d<1$$ Edit: $cn$ on one side only. What I need to ...