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

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Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
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0answers
27 views

Comparing Generalized Continued Fractions

Gosper lays out a method (under Approximations) for comparing regular (a.k.a simple) continued fractions which have all partial numerators set to 1. Continue comparing terms until they differ, then ...
3
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1answer
226 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 ...
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0answers
28 views

Question about set and series notation in the GSP Algorithm

I tried this in programming forums with no luck and since my primary issue is with the notation, I'm asking here: I have the definition of the GSP Algorithm: Definition However I don't understand ...
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0answers
32 views

Produce a list of the most-similar units, given various correlations/relationships

I have a database full of units (U1 - U50, U51...) where every unit has the same standard attributes (A1 - A10) and where a % of each attribute defines the amount of that attribute for that particular ...
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1answer
61 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
3
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1answer
87 views

Limit of a recursiv trigonometric function

So while doing my math homework I recently stumbled upon this little thing: It seems that $y_n = \sin(y_{n-1}) + k$ with arbitrary $y_0$ and $k$ converges to a definite value for $n \to \infty $. ...
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2answers
85 views

Wine problem - ratio and mixture

Question $8$ litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the ...
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1answer
55 views

Understand and an algorythm to Maximize number of triangles from a set of points on XY plane

Given: Set of points (x, y) Looking to: Maximize count of triangles that can be formed. Each triangle which is enclosed within another (with/without shared edge) will be counted again. Specifics on ...
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52 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
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0answers
98 views

Building Minimum warehouses

A big international retailer is setting up shop in India and plans to open stores in N towns (3 ≤ N ≤ 1000), denoted by 1, 2, . . . , N. There are direct routes connecting M pairs among these towns. ...
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1answer
120 views

How to prove a very basic algorithm by induction

I just studied proofs by induction on a math book here and everything is neat and funny: the general strategy is to assume the LHS to be true, and use it to prove the RHS (for the inductive step). Now ...
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2answers
32 views

Finding the inhomogeneous solution

$x_{n+2} = x_{n+1} + 20x_n + n^2 + 5^n \text{ with } x_0 = 0 \text{ and } x_1 = 0$ How would I find the inhomogeneous solution for this since the homogenous solution is 0 given initial conditions?
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2answers
69 views

Towers of Hanoi recurrence relation

How would I do this recurrence relation?
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2answers
49 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 ...
2
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1answer
69 views

Algorithm - iterative method

I'm stuck on an exercise on algorithms, can you help me with this exercise? Solve this recursion using iterative method. $$T(n) = \begin{cases}1 & n=1;\\ 2, & n=2;\\ T(n-2) + n/2,& ...
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1answer
26 views

recursive relation for putting signs in 2*n table

Consider we have a $2\times n$ table and we want to put a sign in some of the cells, and we don't put signs in both adjacent cells give a recursive phrase that shows that how many ways we can do that? ...
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1answer
34 views
0
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1answer
44 views

Number Triangle pattern

I have a number triangle as follows: $$\begin{array}{|c|c|c|} \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ \hline 0 ...
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1answer
33 views

Why $T(n) = 2T(n-1) + O(1)$ is $\Omega(2^n)$?

I was told that the complexity of $T(n) = 2T(n-1) + O(1)$ is $\Omega(2^n)$; however, since I was not convinced, I searched in the Internet and all I found is that problem or very similar ones with ...
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1answer
63 views

N balls of k colors on a cirlce, no two neighboring balls have same color - recursive algorithm

Suppose we have N balls of k different colors. What is the number of arrangements of these N balls on a circle with no two neighboring balls having the same color? Actually, the task is to make up a ...
2
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2answers
74 views

Master theorem - why the log factor?

I think I finally managed to fully understand the master theorem but there's one thing left in the second clause (I'm following here: ...
4
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1answer
58 views

Problem understanding Master theorem

I'm studying the Master theorem (for the analysis of recursive algorithms) and I perfectly understand why a binary search is of order $\log_2(n)$. I also understand that if we formulate it as $T(n) ≤ ...
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1answer
75 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 ...
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1answer
82 views

Karatsuba Method

For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}-1$, how do I check that multiplying $f(x)$ and $g(x)$ by the Karatsuba method requires $3^{r-1}$ multiplications in the field $F$?
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1answer
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How would I go about converting $U(n)= 4^n+U(n-1)$ into an explicit form? [closed]

I have the recursive function $U(n)= 4^n+U(n-1)$, and I'd like to convert it into an explicit form. If you could also walk me through the process that would be great. Thanks!
2
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2answers
84 views

Master theorem, algorithms $T(n) = 2T(n/3) + \log n$

Can I solve $ T(n) = 2T(n/3) + \log n $ using the master theorem?It doesn't seem to fit in any case.
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2answers
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Relation between series and equations

There is following quotes from wiki on Plastic number: The powers of the plastic number $A(n) = ρ^n$ satisfy the recurrence relation $A(n) = A(n − 2) + A(n − 3)$ for $n > 2$. And 2nd is that ...
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1answer
58 views

Find a linear-time algorithm for finding if element occurs n/4 times

Give a linear-time algorithm that determines whether an unsorted sequence of n real numbers contains a number that occurs at least n/4 times in the sequence. You algorithm should report “no” if ...
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1answer
48 views

Find a recursive algorithm to find $a^{2^n}$

Edit1: Used Latex. =] Edit2: Thanks for the guidance to the users below. Really helped me out editing the post and guidance on the math problem. The question gave me a hint: $a^{2^{n+1}} = (a^{2^n}) ...
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6answers
758 views

Non-literal applications of “Shortest Path” algorithm?

It's obvious that it's used in stuff like Google Maps, but what are some more metaphorical applications where you're minimizing the path between nodes (which can represent anything)
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1answer
55 views

How should I proceed to solve this recurrence relation: $T(n) = T(n - 1)^2$

I tried to solve this recurrence relation, but I was confused when I had to determine the pattern. $$ T(n) = \begin{cases} 3, & \text{if }n = 0 \\ T(n - 1)^2, & \text{if }n > 0 \end{cases} ...
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0answers
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Chaining recursive functions

$T(0,r) = k$ $T(n,r) = k + T(n-1,r) + T(n-1,r+1)$ $T(n,0) = k + T(n-1,0) + T(n-1,1)$ $T(n,1) = k + T(n-1,1) + T(n-1,2)$ $T(n,2) = k + T(n-1,2)$ $T(n,3) = k$ Compact: $T(n,0) = k' + T(n-1,0) + ...
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1answer
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$T(n) = 2T(n/2) +n\log n$ - Algorithms

According to this http://en.wikipedia.org/wiki/Introduction_to_Algorithms $T(n)=2T(n/2)+n\log n$ is not case 3 of Master Theorem, can someone explain me why? And which case of master theorem it is?
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1answer
372 views

Solving the recurrence relation $T(n) = T(n-\sqrt n) + 1$

I have an algorithm that at each step can discard $\lceil\sqrt(n)\rceil$ possibilities at a cost $1$. The solution to the recurrence relation below is related to the question of complexity of such ...
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1answer
28 views

How to calculate rental income of an appreciating property?

I'm trying to calculate the total rental income I might receive over 25 years. The rent is 5% of the property value, which is initially £300k. If the property itself increases in value by 3% per year, ...
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1answer
44 views

how to write a recursive definition

so the question asks to define s(n) as the number of strings of a's b's and c's of length n that do not contains "aa". write a recursive definition for s(n). what is s(0),s(1),s(2),s(3). i had to ...
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1answer
83 views

Combinatorial Proof of Identity

How do I build a combinatorial proof of the following recursion: $$\binom {n}{k} = (k+1)\binom {n-1}{k}+(n-k)\binom {n-1}{k-1}$$ I'm having really big difficulties in finding the right way to ...
3
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1answer
44 views

$n$th term of a recursive formula

I have a formula $$ 1 + px + \dfrac{p(p-1)}{1*2}x^2 + \dfrac{p(p-1)(p-2)}{1*2*3} x^3 $$ can someone please tell me what the formula is for the $n$th term of this recursive definition ? Do I have to do ...
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0answers
57 views

Proof relating to Euclidean Algorithm

The question is as follows: (1): Let m and n be positive integers with n < m and let r be the remainder when m is divided by n. Prove that $$r < \frac m2$$ (2): The Euclidean Algorithm for ...
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0answers
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Solving a Recurrence Relation With Summation and Tau Function

How can I solve the following: $$ T(n) = \sum_{i = 1}^{d(n) - 2}T(v_i) + \sum_{i = d(n) - 1}^{n - 1}c + c' $$ Where $d(n)$ is the Tau function, and v is the set of values dividing n. e.g. $d(18) = ...
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0answers
43 views

Verify that this recurrence relation is in O(log n)

For the recurrence $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ is in $\Theta(lg n)$ My attempt at a solution (mostly just wanting to verify its correct). Lower Bound: $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ ...
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2answers
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Solving this recurrence relation

Hi all I'm preparing for a midterm and the following appeared as a practice problem that I'm not quite sure how to solve. It asks to find a tight bound on the recurrence using induction $$ {\rm ...
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What is this method of dividing a plane called?

I have an idea of a method for recursively dividing a plane, and as I'd like to do more research about this algorithm and the set of points that it produces, I'd like to know what it's formally known ...
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1answer
297 views

Cyclic tower of hanoi problem [duplicate]

If I have 3 rods in a circle and it is allowed to move the disks only in the clockwise direction. How many moves is necessary to move n disks from first rod to the third rod?
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2answers
54 views

Calculate a recursive equation in terms of theta

I am struggling with the following equation for one week! Please help me solve it. $$T(n)=T(\frac{n}{2})+\frac{n}{logn}$$ So far, I have come to the equation $T(n)=\Sigma \frac{2^x}{x}$
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1answer
134 views

Quadratic Map Solution

I read this on Wolfram Alpha. It states that: a quadratic recurrence relation uses a second degree polynomial to express $x_{n+1}$ as a function of $x_n$. A "quadratic map", then, is a recurrence ...
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1answer
387 views

How to solve this recurrence relation with Sigma notation (f(n, m) = f(n - 1, m) + f(n, m- 1) + c?

This recurrence relation was inferred from the function $f(n, m) = f(n - 1, m) + f(n, m-1) + c$. After expanding the latter, I ended up with the following: $$f(n,m)=\begin{cases} 0,&\text{if ...
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1answer
65 views

Transfer a program into a logarithm formula

I have been thinking what is the math formula for the piece of code I wrote below. Assume there are B, D and F three positive integers, and D is smaller than F. Given D and F, the value of B would be ...
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Recursive and Recusively enumerable

{1^n | n finite integer and >1}, is this language recursive? I'm not sure how to prove that a language is recursive, I only know that there should be a TM that accepts any finite input and then ...