Questions dealing with recursive algorithms. Their analysis often involves recurrence relations, which have their own tag.

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14
votes
6answers
2k views

How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$

How to solve this particular recurrence relation ? $$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$ such that $f_2 = 12, f_3 = 24$ and so on. I tried out a lot but due to $(-1)^n$ I am not able to ...
7
votes
2answers
924 views

unorthodox solution of a special case of the master theorem

I am asking for references regarding a special case of the master theorem. This theorem seems to appear quite a lot on this site, prompting me to study it in more detail, e.g. see my posts here and ...
8
votes
3answers
2k views

Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?

1) Solve the recurrence relation $$T(n)=\begin{cases} 2T(n-1)+1,&\text{if }n>1\\ 1,&\text{if }n=1\;. \end{cases}$$ 2) Name a problem that also has such a recurrence relation. The ...
3
votes
3answers
463 views

Recurrence relation by substitution

I have an exercise where I need to prove by using the substitution method the following $$T(n) = 4T(n/3)+n = \Theta(n^{\log_3 4})$$ using as guess like the one below will fail, I cannot see why, ...
0
votes
3answers
899 views

Counting arrays with gcd 1

I want to calculate the number of arrays of size $N$, such that for each of it's element $A_i, 1 \leq A_i \leq M$ holds, and gcd of elements of array is 1. Constraints: $1 \leq A_i \leq M$ and $A_i$ ...
1
vote
2answers
330 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 ...
0
votes
3answers
127 views

How can $n \lg n = O(n^{log_3 4 - r})$?

How can I understand this bound, for me it is not true. $$n\lg n = O(n^{\log_3 4 - r})$$ where $\lg n = \log_2 n$ and $r > 0$ I'm trying to solve this recurrence $T(n) = 4T(n/3) + n\lg n$ using ...
4
votes
1answer
321 views

Computing sums of divisors in $O(\sqrt n)$ time?

I have a sequence: $1,3,5,8,10,14,16,20,23,27,\dots$ I know that the recursive relation is: $$p[i] := p[i-1] + \text{number of factors of $i$}, \quad \text{with $p[1]=1$.}$$ How do I solve this ...
0
votes
2answers
104 views

Explanation needed on this rather basic recurrence solution

We are studying about recurrences in our analysis of algorithms class. As an example of the substitution method (with induction) we are given the following: $$T(n) = \lbrace 2T\left(\frac{n}{2}\right) ...
0
votes
2answers
4k views

Recurrence telescoping $T(n) = T(n-1) + 1/n$ and $T(n) = T(n-1) + \log n$

I am trying to solve the following recurrence relations using telescoping. How would I go about doing it? $T(n) = T(n-1) + 1/n$ $T(n) = T(n-1) + \log n$ thanks
8
votes
2answers
1k views

Proof of clockwise towers of Hanoi variant recursive solution

This is from one of the exercises in "Concrete Mathematics", and is something I'm doing privately, not homework. This is a variant on the classic towers of Hanoi, where all moves must be made ...
4
votes
1answer
243 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
6
votes
3answers
388 views

Recurrence with varying coefficient

Problem 1 $$ {\rm f}\left(n\right) = \frac{1}{n}\, \left[{\rm f}\left(n - 1\right)k_{0} + {\rm f}\left(n-2\right)k_{1}\right]\tag{1} $$ ( This can also be written as ${\rm Q}\left(n\right) = ...
4
votes
1answer
1k views

Stepping through the Josephus problem

I'm trying to figure out how the math in the Josephus problem works exactly. I first heard of it for programming purposes so that's where my focus has been. The formula does seem to be recursive from ...
2
votes
2answers
585 views

Solving recurrences with boundary conditions

I'm trying to follow CLRS ("Introduction to Algorithms") and I just hit a question in a practice assignment I found online that I just can't make any sense of. Consider this problem: Show that ...
4
votes
2answers
6k views

Proof by the substitution method that if $T(n) = T(n - 1) + \Theta(n)$ then $T(n)=\Theta(n^2)$

How to prove by the substitution method that if $T(n) = T(n - 1) + \Theta(n)$ then $T(n)=\Theta(n^2)$? I've tried the following and got stuck $$ \begin{align} T(n) &= T(n - 1) + \Theta(n) \\ ...
2
votes
3answers
794 views

Prove algorithm correctness

I'm wondering if there exists any rule/scheme of proceeding with proving algorithm correctness? For example we have a function $F$ defined on the natural numbers and defined below: ...
1
vote
1answer
77 views

Does this sequence of operators in Hilbert space, given by an algorithm, terminate

Let $H$ be an infinitedimensional Hilbert space and $T$ a compact selfadjoint operator in it. Consider the following Algorithm: Let $$ H_{1}=H,\ T_{1}=T $$ and let $\lambda_{1}$ be that ...
0
votes
6answers
328 views

Obtain the formula for the following sequence

I can't seem to figure out how to find an algebraic formula for the following sequence of numbers. $$0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0$$ Can somebody ...
8
votes
1answer
157 views

What are the solutions for $a(n)$ and $b(n)$ when $a(n+1)=a(n)b(n)$ and $b(n+1)=a(n)+b(n)$?

If you have the following recurrence relations : $a(n+1)= a(n) b(n)$ $b(n+1)= a(n) + b(n) $ How do you find the form of $a(n)$ and $b(n)$ ? I suspect there isn't a closed form , but a infinit sum ...
8
votes
1answer
777 views

Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?

I have been reading Introduction to Algorithms by Cormen. Before explaining Strassen algorithm the book says this: Strassen’s algorithm is not at all obvious. (This might be the biggest ...
7
votes
1answer
250 views

Finding $a_n$ for very large $n$ where $a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $

I have a recurrence relation, $$ a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $$ for $n>3$ and $a_1 = 0, a_2 = 0, a_3 = 1$ I have to find the value of $a_n$ for very large values of n. I tried ...
5
votes
4answers
2k views

A 3rd grade math problem: fill in blanks with numbers to obtain a valid equation

Even though this is a 3rd-grade math problem, people found it extremely hard. Any people have a solution, or algorithm is welcome. I'll try make a program base on the algorithm and see if it's ...
5
votes
1answer
1k views

Karatsuba multiplication with integers of size 3

I understand how to apply Karatsuba multiplication in 2 digit integers. $$\begin{array} \quad & \quad & x & y \\ \times & \quad & z & w \\ \hline \quad ...
7
votes
1answer
160 views

Another recurrence… $T(n)=\sqrt{2n} \cdot T(\sqrt{2n}) + \sqrt{n}$

I'm trying to solve the following recurrence : $$T(n)=\sqrt{2n}\cdot T(\sqrt{2n}) + \sqrt{n}$$ I've tried substituting $n$ for some other variables to transform the above to something easier without ...
6
votes
1answer
557 views

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is an exercise from Chapter 1 of "Concrete Mathematics". It concerns the Towers of Hanoi. Are there any starting and ending configurations of $n$ disks on three pegs that are more than $2^n - ...
5
votes
1answer
183 views

Strange Recurrence: What is it asymptotic to?

So I have the following recurrence relation for the growth rate of an algorithm: $T(n)$ = time taken by algorithm to solve problem of size n: $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$ Clearly this ...
2
votes
2answers
1k views

Recurrence Relation for Strassen

I'm trying to solve the following recurrence relation (Strassen's):- $$ T(n) =\begin{cases} 7T(n/2) + 18n^2 & \text{if } n > 2\\ 1 & \text{if } n \leq 2 \end{cases} ...
1
vote
1answer
68 views

Recurrence relation using substitution method

How do I solve the following recurrence using substitution method? $$T(n) = T(n-1)+C$$ I've found reference to so many examples on line but most of the examples are of the form $$T(n) = ...
1
vote
1answer
88 views

Recursive Function - $f(n)=f(an)+f(bn)+n$

I've got this recursive function $f(n)=f(an)+f(bn)+n$, and I need to find $θ$ on $f(n)$, as $a+b>1$. Using a recursive tree, I managed to bound it by $n\log(n)$ from the bottom and by $n^2$ from ...
-1
votes
2answers
52 views

Help solving recursive relations

$x_{n+2} = x_{n+1} + 20x_n + n^2 + 5^n \text{ with } x_0 = 0 \text{ and } x_1 = 0$ How would you solve this recursive relation? I have the homogenous solution, but am having issues with the ...
8
votes
2answers
2k views

Numerical method for finding the square-root.

I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 ...
7
votes
3answers
407 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
2
votes
3answers
8k views

Worst case analysis of MAX-HEAPIFY procedure .

From CLRS book for MAX-HEAPIFY procedure : The children's subtrees each have size at most 2n/3 - the worst case occurs when the last row of the tree is exactly half full I fail to see this ...
4
votes
1answer
48 views

Recursive Sum of Previous Term and its Inverse

Can anyone help me with finding a closed form for $F_n$ where $$F_0=x_0$$ $$F_{n+1}=F_n+\frac{1}{F_n}=\frac{F_n^2+1}{F_n}$$ I could imagine this already having been done, in which case I'd appreciate ...
8
votes
3answers
771 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
5
votes
0answers
124 views

Is there a simplification for the coefficients generated with the Mandelbrot iteration rule?

The Mandelbrot Set is obtained using the equation $z_n=z_{n-1}^2+c$ for some constant $c \in \mathbb{C}$ with $z_0=0$. Therefore, $z_1=c$, $z_2=c^2+c$, $z_3=c^4+2c^3+c^2+c$, etc. I have a function ...
2
votes
0answers
119 views

Finding a solution to $\sum _{n=1}^{n=k} \frac{1}{n^x}+\sum _{n=1}^{n=k} \frac{1}{n^y}=0$

Scroll down to the update to see what I am meaning. The Mathematica program below finds a solution to the equation: $$\sum _{n=1}^5 \frac{1}{n^x}+\sum _{n=1}^5 \frac{1}{n^y}=0$$ My question is if you ...
1
vote
1answer
106 views

Find a Recurrence Relation

I want to find a recurrence relation for number of decimal numbers with length n, (we called $a_0$ ) that not includes 0 and any combination of 11,12, 21. i see the result is: ...
0
votes
0answers
48 views

Can linear execution time be achieved [duplicate]

The SELECT algorithm determines the $i$th smallest of an input array of $n>1$ distinct elements by executing the following steps. Divide the $n$ elements of the input array into $\lfloor ...
0
votes
1answer
695 views

Interesting Recurrence Relation $T(n) = T(\sqrt{n}) + T(n-\sqrt{n}) + n$

I found an interesting recurrence that I do not know how to solve. I think this has to do with quicksort with pivots at rank $\sqrt{n}$. I do not know how to approach this problem nor found any ...
5
votes
2answers
304 views

Sum of the series formula

I need to figure out the sum of the series as quickly as possible in a program given n and k: $$f(n,k)= ...
5
votes
1answer
216 views

Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$

I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $ then the general equation looks like the following: $$ F_n = F_{n-1}(F_{n-2}) + F_{n-3} $$ So far I have got the ...
5
votes
1answer
776 views

Applying a Math Formula in a more elegant way (maybe a recursive call would do the trick)

This is my first post in Math section. I've been redirected here from StackOverflow, users from there suggested me to ask here. This could appear a little bit complex at first sight but afterall ...
5
votes
1answer
278 views

Nesting functions to understand nesting series

So I am working on a problem that involves an expression that has a "nested" sigma notation. Maybe "nested" isn't the correct word, because you start with an input (of your choosing) and the output ...
3
votes
1answer
227 views

number of derangements

In the normal derangement problem we have to count the number of derangement when each counter has just one correct house,what if some counters have shared houses. A derangement of n numbers is a ...
2
votes
5answers
162 views

How to give a good guess to the recurrence relation problem [duplicate]

I have been trying to solve the following recurrence relation $$T(n)=2T(\frac{n}{2}) + nlgn$$ by using substitution method. I started to compute $T(1)$ ,$T(4)$,$T(8)$,$T(16)$ to guess a solution as ...
2
votes
0answers
240 views

Algorithm for reversion of power series?

Given a function $f(x)$ of the form: $$f(x) = x/(a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+...a_nx^n)$$ Let $A$ be an arbitrary (any) infinite lower triangular matrix with ones in the diagonal: $$A ...
1
vote
1answer
58 views

Prove by induction T(n) = T(⌊n/2⌋) +T(⌊7n/16⌋) + n

Prove by induction on n that T(n)=O(n), where T(0)=1, T(n) = T(⌊n/2⌋) +T(⌊7n/16⌋) + n So far I have, Base Case: n = 1 [1/2] + [7/16] + 1 T(1) = 1 Induction hypothesis: Assume that for arbitrary ...
1
vote
1answer
145 views

What's a useful recurrence relation for $(2n)!$ in terms of $n!$?

I want to make an algorithm for $n!$ which will divide the number in half and call the algorithm again until n is so close to $0$ that a value of $1$ can be safely returned, and use the value of each ...