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Questions tagged [recurrence-relation]

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

2
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2answers
28 views

Finding recurrence relation for running time of an algorithm

I am pretty new to this, consider the following algorithm: ...
0
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0answers
18 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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2answers
63 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
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0answers
76 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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2answers
52 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
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0answers
44 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) + ...
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0answers
116 views

Solving Recurrence

Given the recurrence $\forall \{a_i\}\subseteq \mathbb N.\ \ T((\sum_i{a_i})!)\leq \sum_i {T(a_i!)+T((\sum_i {a_i})!-\prod_i (a_i!))}$ with $T(O(1))=O(1) $ How should I solve this? I tried to ...
1
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3answers
62 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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3answers
99 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
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2answers
24 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
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4answers
105 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
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1answer
44 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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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
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1answer
28 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
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2answers
77 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
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0answers
31 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
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1answer
37 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
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1answer
43 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
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1answer
59 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
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2answers
140 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
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1answer
31 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. ...
1
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1answer
32 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 ...
0
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1answer
26 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. ...
1
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1answer
37 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 ...
1
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1answer
50 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 ...
0
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0answers
35 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 ...
2
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1answer
89 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....
2
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2answers
273 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) = ...
3
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1answer
50 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 ...
2
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3answers
183 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)...
1
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1answer
59 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 $T(n)=aT(n/b)+f(n)$ ...
1
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0answers
26 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[...
2
votes
1answer
48 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 ...
1
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1answer
119 views

How to solve F(n)=F(n-1)+F(n-2)+f(n) recursive function?

Like in the title the following equation: F(n)=F(n-1)+F(n-2)+f(n) F(0)=0, F(1)=1 ...
1
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1answer
105 views

DP recurrence relations: Coin change vs Knapsack

Take: KP recurrence relation $ max { [v + f(k-1,g-w ), f(k-1,g)] } $ if w <= g and k>0 CCP recurrence relation $ min {[1 + f(r,c-v), f(r-1,c)]} $ if v <= c and r>0 I don't ...
1
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0answers
30 views

Trouble with Master Theorem concerning logarithm and square root [duplicate]

I have trouble understanding how to apply the master theorem in the following problem: $$T_2(1) = 1; T_2(n) = 4T_2(2^{\log \lfloor \frac{n}{2}\rfloor}) + \sqrt{n} \text{ for } n > 1.$$ My ...
0
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2answers
58 views

How can I solve the recurrence $f(n) = 3f(\frac{n}{4}) + \log(n)$?

The master theorem didn't work here. I tried to do the substitution method but I ended up with an additional term: $2Σ(i \cdot 3^i)$. Also I should find the solution $g(n)$ such as $f=\Theta(g)$.
1
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1answer
67 views

What is the solution of $T(n, m) = T(n, m-1) + T(n-1, m) + c$?

Consider the recurrence $$ T(n,m) = T(n,m-1) + T(n-1,m) + c, $$ with base cases $T(n,0) = T(0,m) = 1$. This is the complexity of a recursive algorithm for the longest common subsequence, I know that ...
1
vote
2answers
80 views

How to compute the complexity of $T(n) = T(n-2)+T(n-3)+2T(n/3)$?

$T(n) = T(n-2)+T(n-3)+2T(n/3)$ and $T(n)=1$ for $n<4$. I tried to compute the complexity of $T(n) = T(n-2)+T(n-3)+2T(n/3)$ using the recursion tree but it's not clear enough for me to make a guess ...
2
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2answers
48 views

Possible to use Master theorem? $T(n) = aT(\lfloor \frac{n}{b} \rfloor) + g(n)$

The master theorem can be used in case of a recurrence relation of the form $T(n) = aT(\frac{n}{b}) + g(n)$ But is it possible to use the master theorem for recurrence relations of the form $T(n) = ...
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votes
1answer
211 views

How to solve $T(n) = 3 T(n-1) + 10 T(n-2) + 7 \cdot 5 ^ n$?

Consider the recurrence $$ T(n) = 3 T(n-1) + 10 T(n-2) + 7 \cdot 5 ^ n, $$ with base cases $T(0) = 4$ and $T(1) = 3$. How do I solve such a recurrence?
1
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1answer
47 views

Find an asymptotic bound for $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+…+T(\frac{n}{2^k})$

Given is the following recurrence relation: $T(n)=n^2+T(\frac{n}{2})+T(\frac{n}{4})+T(\frac{n}{8})+...+T(\frac{n}{2^k})$ where $k$ is some constant and $n = 2^t$ for some $t\in \mathbb{Z}$. I'm ...
1
vote
2answers
68 views

Deriving the average depth for a randomly generated binary search tree

If $D(n)$ is the internal path length (sum of the depths of all nodes) for some tree $T$ with $n$ nodes then we have the following recurrence relation: $$D(n)=D(i)+D(n-i-1)+N-1$$ where I simply taken ...
0
votes
1answer
134 views

Coin Change Problem Recurrence Relation with one parameter

I have been looking through the recursive formulation for the coin change problem here and am wondering if it is possible to define the function $ C(N, m) $ in one parameter as $C(N)$, therefore not ...
-1
votes
2answers
45 views

Using Iterative method to find recurrence relation vs Master Theorm

I'm trying to solve this recurrence relation using the iterative method and i keep getting the different answer from using the master theorem. $$\begin{aligned} T(n) &= 5T(n/2) +n^2 \\ &=...
1
vote
2answers
422 views

Solving T(n)=T(n−1)+2T(n−2) using substitution

I am trying to solve the following Recurrence relation using substitution method and I am stuck almost half way. I know the answer is 2^n but I can't reach it. At first, my question is: Who decicdes ...
1
vote
1answer
39 views

Big-Oh vs Theta in recurrence tree method

I am solving this problem from here. The given relation is $$T(n) = 2 T(\frac{n}{2}) + n^2, \, T(1) = 1$$ The solution via recurrence tree method is given as: The zeroth level has a single node ...
1
vote
2answers
69 views

Big-O Solving Recurrence Relation by iteration with fractions

I was trying to solve the recurrence relation in order to get a some big-O bound $$ B(n) = B(n-4) + \frac{1}{n} + \frac{5}{n^{2} + 6} + \frac{7n^{2}}{3n^{3} + 8}$$ by following the accepted answer ...
0
votes
2answers
24 views

Which one is the correct way to change functional notation while solving recurrence relation?

In CLRS 3rd edition P. No 86, the relation $T(2^{n}) = 2T(2^{n/2}) + n$ is changed to $S(n) = 2S(n/2) + n$ My first question is, why we can change any recurrence relation like this? And, ...
1
vote
1answer
39 views

Optimizing the problem

I have a recurrence relation: $$f(a,b) = \begin{cases} 1 & (a,b) = (0, 0)\\ 1 & (a,b) = (a, 0)\\ 0 & (a,b) = (0, b)\\ 2a & (a,b) = (a,1)\\ f(a-1,b) + f(a-2, b-1) + f(a-1,b-1)...