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

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Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
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174 views

Fractional iteration of the Newton-approximation-formula: how to resolve one unknown parameter?

For my own exercising I tried to find a closed form expression for the Newton-approximation algorithm, beginning with the simple example for getting the squareroot of some given $ \small z^2 $ by ...
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41 views

Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers ...
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40 views

What's the next “recursion” here?

Plotting a single 3d helix is x = cos(t); y = sin(t); z = t; From this equation: x = [R + a cos(\omega t)] cos t y = [R + a cos(\omega t)] sin t z = h t + a sin(omega t) Comes the awesome ...
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89 views

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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252 views

Question about recursive defined functions.

This question is about counting functions. With counting functions $F$ I mean functions from the positive integers to the positive integers that are strictly nondecreasing and can grow no faster than ...
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410 views

When do floors and ceilings matter while solving recurrences?

I came across places where floors and ceilings are neglected while solving recurrences. Example from CLRS (chapter 4, pg.83) where floor is neglected: Here (pg.2, exercise 4.1–1) is an example ...
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92 views

How can we find the elements?

I want to describe an algorithm with time complexity $O(m)$ that, given a set $M$ with $m$ numbers and a positive integer $p \leq m$, returns the $p$ closest numbers to the median element of the set ...
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40 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 $z_0=c$. Therefore, $z_1=c^2+c$, $z_2=c^4+2c^3+c^2+c$, etc. I have a function $f(n,x)$ ...
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71 views

Local informatics Olympiad and Algorithm

I see one of recent local informatics Olympiad question. i have a trouble to solve it. any idea? hint? or solutions? thanks to all creative man. We have two function $P_1, P_2$ and input an array $n$ ...
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88 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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52 views

Proof relating to Euclidian 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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38 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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30 views

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 ...
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150 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 ...
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83 views

Interesting Recursive Continued Fraction Limit

I was messing around with recursive functions the other day and came up with something that could be interesting: Definition of $\bar{\Xi}(n)$:\ Let $\Xi ...
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59 views

Difference Sets

suppose we have a set $$P=\{p_1,p_2,...,p_K\}$$ where $$1\leq p_k\leq N , k=1,...,K \qquad \& \quad p_k \in \mathbb{N} $$ and $p_k$'s are distinct. We calculate the differences as: $$d=p_i-p_j\mod ...
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142 views

Subtrees of a tree

I have a given tree with n nodes. The task is to find the number of subtrees of the given tree with outgoing edges to its complement less than or equal to a given number K. for example: If ...
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126 views

Recursive Sequence from Finite Sequences

I'm searching for the name of these kind of sequences: Suppose you start off with a finite sequence containing one term: S0 = {3} To get the next sequence you ...
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104 views

Is there a closed form for this recurrence?

Given $$ E_{n,k} = \begin{cases} 0 & \text{ if } n \leq k \\ n & \text{ if } k = 0 \\ \sum_{i=0}^{n-1} \dfrac{1}{n} \cdot E_{i,k-1} & \text{ otherwise } \end{cases} $$ I wonder is there ...
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75 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
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76 views

Proving that an effective procedure is correct

I will start with definitions, theorems, and a few solved exercises which I am taking as theorems now. My actual question will be last, if you want to scroll ahead to see it. Definitions: (1) The ...
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172 views

google page rank algorithm with range of values

According to the wikipedia article, the iterative google page rank algorithm is defined as follows: Can this be modified to include a range of values, not just binary, and would it look something ...
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130 views

Help on Generating function - Analysis of Median of three - Quick Select

I am trying to find out the average case analysis of median of 3 - quick select. The recurrence relation is \begin{equation} C_{n,j} = 1 + \sum_{k=1}^{j-1}\pi_{n,k}C_{n-k,j-k} + \sum_{k=j+1}^{n} ...
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184 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times ...
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26 views

Recursion tree T(n) = T(n/3) + T(2n/3) + cn

I have a task: Explain that by using recursion tree that solution for: $T(n)=T(\frac n3)+T(\frac 2n3)+cn$ Where c is constance, is $\Omega(n\lg n)$ My solution: 1. Recursion tree for $T(n)=T(\frac ...
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5 views

algorithm related to dyadic decomposition

Starting from an integer $n >0$, iterate the two operations of substracting one and dividing by $2$ (when the number is even) until $1$ is reached. Thus when the number is odd we can only substract ...
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11 views

Using FFT to compute DFT of a polynomial

i currently studying about FFT and DFT and we were given simple question: Use the recursive FFT to compute the DFT of this polynomial of '3' degree: $$-1\:+\:4x\:+\:3x^2$$. So, i go to this ...
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19 views

Closest pair algorithm in high dimension?

2D case is clear. But with dimensions higher than 2 I should choose a special partitioning hyperplane for the divide and conquer algorithm to get O(n log n). I am confused because to choose this ...
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57 views

Help me generalize what this divisor transform does.

I have an algorithm which takes as input the series expansion of: $$\frac{-(1 + ax(-2 + x + ax))}{-1 + ax} \tag 1$$ or expressed differently: ...
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52 views

Trace and transpose of a Matrix

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1 =sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2} =\frac{s}{n+2}\{ R_{n+1} ...
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25 views

Solution of ODE with initial values

Given data in the problem ${\psi'(t)}_{3 \times 3}=A_{3 \times 3}\psi(t)_{3 \times 3}, \psi(0)_{3 \times 3}=R^{cl}_{3 \times 3} \\ \phi'(t)_{3 \times 3}=t\hspace{.1cm}B_{3 \times 3} \phi(t)_{3 ...
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Finding the explicit form of the recursive function $P_{1(n)}=\left\lceil\frac12P_{1(n-1)}\right\rceil+\left\lfloor\frac12P_{2(n-1)}\right\rfloor$

I'm trying to find the explicit form of the recursive function $$P_{1(n)}=\left\lceil\frac12P_{1(n-1)}\right\rceil+\left\lfloor\frac12P_{2(n-1)}\right\rfloor\;.$$ First, let me explain what this ...
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34 views

Recurrence Derivative

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ ...
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33 views

Need help checking my recurrence for a simple algorithm

All I'm writing to get a second opinion on the algorithm shown in this link. I'm pretty sure its supposed to be $T(n)=2T(n/2)+n$ but I can't see where I'm supposed to get the +n from. So far I'm ...
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45 views

Using a Recursion Tree to solve the recurrence $T(n) = \sqrt n T(\frac{n}{2}) + 10n$?

I am attempting to solve the above recurence by giving tight $\Theta$ bounds. Assume that the logs here are all base 2! To solve a recursion tree as far as I understand, I need two things. The ...
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54 views

Algorithms recurrence equation using unwinding / substitution

This is homework for introduction to Algorithms this is what I have below. (25 pts) Use the direct unwinding to prove that the recurrence equation T(n)= 2T(n/2)+ nlogn has a lower bound of ...
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35 views

Divisible 4 for every different number?

My formula is for ABC 3 digits number; 100A + x*B + y*C. What should coefficient of B and C be for everytime different result for different number? (Result number ) mod 4 has to be zero. For ...
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53 views

Solving the recursion $F(n)=K_0F(n-1)/(n-1)+K_1F(n-2)/(n-2)$

Please help me in solving the recursion $F(n)=K_0\frac{F(n-1)}{n-1}+K_1\frac{F(n-2)}{n-2}$, preferably using power series for the values of $F(n)$ in terms of $n$. Here $K_1$ and $K_2$ are ...
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50 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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51 views

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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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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28 views

Determining the optimally scoring move on a probabilistically represented 2D grid in real time

I'm posting this to StackOverflow, cstheory.stackexchange.com, and math.stackexchange.com because I'm not really sure where it fits best. I hope that's OK. I have a 2D grid (size varies per map, ...
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25 views

Is there a method to calculate Mahalanobis distance incrementally?

By incrementally (or recursively) I mean to update the pool/group of values/vectors as soon as more vectors are available without having to recompute the entire covariance and inverse matrices. I ...
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55 views

Is undecidability of arithmetic a corollary of Tarski undefinability theorem?

Arithmetic is undecidable, in other words the set of Godel numbers of theorems of arithmetic is not recursive, and so there is no algorithm/ recursive function to decide if a statement is provable or ...
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154 views

Sequential Algorithm to generate Fractal (Koch's snowflake)

As part of an assignment I had developed a sequential algorithm to generate a Koch's snowflake. Algorithm I have encountered so far have been recursive and iterations generate closer approximations. ...
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132 views

How to compute this recursion in linear time?

Can the following iterative update on a $n$-element vector $\mathbf{x}_t$ be computed in $O(n)$ computations? \begin{align*} \mathbf{x}_{t+1} & = a_t\mathbf{y}_t + \mathbf{A}_t \mathbf{x}_t \,,\\ ...
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50 views

How to derive recursive equation for expected discounted utility function?

I am trying to mathematically represent expected discounted utilities when the length of a lifetime is uncertain. $V_0$ is the expected discounted utility at $t=0$ and can be represented as such: ...
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211 views

Help with find recurrence relation running time.

Write a recurrence relation describing the worst case running time of each of the following algorithms, and determine the asymptotic complexity of the function defined by the recurrence relation. ...
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229 views

Recursive number digits power n sum => is there a limit of unique result numbers found?

Say you have a number xyz and you choose to split it to digits, take a power of each digit to three and summarize them. Some numbers gives same result than the original number, for example: 153 = 1^3 ...