# Questions tagged [recurrence-relation]

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

724 questions
Filter by
Sorted by
Tagged with
25 views

### Determine if all the continuous subsequences of an array contain at least one unique element in O(n lgn)

Given an array of length n, how to determine if all the continuous subsequence of this array contains at least one unique element. Any subarray array[start, end] ...
1 vote
57 views

### Solve the recurrence $T(n)=T(n-2)+\frac{1}{\lg{n}}$

Assume this recurrence: $$T(n)=T(n-2)+\frac{1}{\lg{n}}$$ I tried to draw its recurrence tree and I reached that the whole cost is $\dfrac{1}{\lg{n}}+\dfrac{1}{\lg{n-2}}+\dots+\dfrac{1}{x}$ that $x$ is ...
1 vote
29 views

1 vote
30 views

### Solve a recurrence using Akra-Bazzi method where $p$ is not integer and integration is not easy

I recently faced this problem in CLRS ed.4 and couldn't find out how to attack it and solve it. Here's the recurrence: $$T(n)=3T(\frac{n}{3})+8T(\frac{n}{4})+\frac{n^2}{\log{n}}$$ Here's what I tried: ...
37 views

### Solving Recurrence Relations with induction

We got the following tasks in our Higher Algorithm class, to repeat our proof techniques from class: Find asymptotic upper bounds (as sharp as possible) for $T(n)$ in each of the following cases, ...
57 views

45 views

### Prove $T(n)=2T(\dfrac{n}{2})+\Theta(n\log{n})=\Theta(n\log^2{n})$ using induction

Please first take a brief look at my previous question. Here I want to do something similar but for $T(n)=2T(\dfrac{n}{2})+\Theta(n\log{n})$. I know the answer is $T(n)=\Theta(n\log^2{n})$ and I want ...
34 views

### 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 ...
54 views

### Find time complexity of $T(n)=3T(n-2)+O(n)$

I try to find the time complexity of following recurrence relation: $$T(n) = 3T(n-2) + O(n)$$ After subtitution,I get: $$T(n)=3^{\frac{n}{2}}T(0)+\sum_{i=0}^{\frac{n}{2}-1}3^iO(n-2i)$$ I wonder if the ...
1 vote
141 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 ...
40 views

### Is my mathematical representation of search in binary search tree correct?

You are given the root of a binary search tree (BST) and an integer val. Find the node in the BST that the node's value equals <...
45 views

75 views

### Recurrence Relation for Longest Increasing Subsequence Problem

I am trying to solve the Longest Increasing Subsequence(LIS) Problem using different OPT Function than the one which normally used. I have been given this question as an extra credit and I have been ...
29 views

### How to solve recurrences of this type?

$T(n) = 2 T(\lceil \frac{2n}{3} \rceil) + T(\lceil \frac{n}{3} \rceil) + O(n log n)$ From the 3-ary recurrence tree, one can say that $T(n) \geq cnlog^{2}n$ for some constant c, using the shortest ...
111 views

### How do I solve this recurrence equation?

I have to express the solution of the recurrence equation T(n) = T(an) + n where a is a constant, 0 < a < 1, in terms of θ using the iteration method. I am unsure of how I calculate the cost of ...
41 views

### A Problem with Solving a Recurrence relation

I Hope someone Can help me with that: $T(1)=2$ $T(n)=\left(T(\frac{n}{2})\right)^2\cdot2^n$ what is the runtime complexity of the algorithm (base 2) Thanks a Lot!
36 views

### Solving a recurrence relation formula with squared

I hope someone can help me with that: $$T(n)=T(2^{\sqrt{\log n}})+1$$ I will be asked to answer what is the runtime complexity of the algorithm. I tried to set m=2^ and still failed.
45 views

142 views

### How to write recursive function in pseudocode for this number $a_n=n!+2^n$

I need to write recursive function in pseudocode for n-th number term of $a_n=n!+2^n$. Whole code should be contained in one function with $n$ as function argument.
52 views

### How to solve this recurrence relation: T(n) = R(n-1) + n log n R(n) = T(n-1) + n^2

How to solve this recurrence relation: T(n) = R(n-1) + n log n R(n) = T(n-1) + n^2
1k views

106 views

### Prove that the total number of parenthesizations of n matrices is Ω(4^n/n^3/2)

Is it possible to prove the total number of parenthesizations of n matrices is Ω(4^n/n^3/2) using the Induction Method? Recurrence formula from CLRS book When n = 1, the sequence consists of just one ...
134 views

### In merge sort, what will be the time complexity if in each recursion, we break the array in two parts of size 1/4 and 3/4 respectively?

Let's say number of elements are a power of 4. Now if we break the array in parts of 1/4 and 3/4, how do we calculate the time complexity in this case?
73 views

### Finding the runtime out of a recursion formula when using divide-and-conquer

In divide-and-conquer, one uses the following formula to find the runtime: $$T(n) = aT(n/b) + f(n).$$ I am confused with the meaning of the constants $a$ and $b$, as well as by the question of how to ...
66 views

### 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
73 views

### Recurence Relation, specifically understanding substitution rule used

This is a pretty vague question and can be applied to many math problems not just recurrence relations. Above I fully understand, setting up the recurrence relation from the algorithm given. And how ...