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

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Reccurence relation

Closed form of $\ nT_n = 3(n-1)T_{n-1} + 1, n \ge 1$ I've tried calculating some terms, and looking it up on wolframalpha, it sais the generating function is $\frac{exp(x)}{1-3x}$. Where do i start? ...
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Proving by induction that a palindrome contains an even number of $b$s and $c$s

Suppose we want to construct palindromes that contain an $aa$ in the middle if the length is even and an $a$ in the middle if the length is odd. I'm trying to prove by induction that all of these ...
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A recursive definition of palidrome over {a,b}

Can someone please explain how this recursive definition produces palindromes over $\{a,b\}$? For example, how can we get the string $abaaaba$ as a valid string? Rule 1: $\epsilon$, $a$, and $b$ are ...
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A Shifted Sorted Array - finding the shift

I was given a problem concerning a sorted array that is shifted by some "number of spaces", k. For example, take the sorted array $1, 2, 3, 4, 5 ,6$ It is shifted 2 spaces, we get: $5, 6, 1, 2, 3 ...
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Internet problem solving contest question

I am trying to solve a problem from the IPSC http://ipsc.ksp.sk/2001/real/problems/f.html It basically asks to compute the following recursion. ...
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Correctness of complexity analysis of recursive algorithm

Given following recursive equation: $$T(n) = T(n-3) + \Theta(1)$$ Is it correct that this equation is O(1)?
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Recursive relation

Sequence defined by recursive relation $a_{n+1}=\alpha a_n +2$ Prove that if $\left|\alpha\right|\lt1,$ the sequence has a limit independent of $a_1$. I have seen this work for the case when ...
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Algorithm to Generate Number with 0 and 9 [duplicate]

You are given an integer N. Can you find the least positive integer X made up of only 9’s and 0’s such that X is a multiple of N? Is there any Algorithm to generate the least number. Thank you.
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Recursion with Random number?

function foo(n) if n = 1 then return 1 else return foo(rand(1, n)) end if end function If ...
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220 views

how to prove the convergence of fixed point iteration algorithm

Please refer to the below algorithm: Above two steps can be rewritten as, \begin{equation} x(k+1)=\arccos\bigg( -\frac{1}{2(Dr^{\frac {|\sin(2x(k)+\theta)|}{M\sin ...
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171 views

Difficult recursion problem

A student can do the things bellow: a. Do his homework in 2 days b. Write a poem in 2 days c. Go on a trip for 2 days d. Study for exams for 1 day e. Play pc games for 1 day A schedule of n days ...
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Recursion equation

$$f_k = \sum\limits_{k = 1}^{k-1} (f_i + f_{k - i}), \text{ when } k \ge 2, \text{ and } f_1 = 1$$ Find/guess a closed form for $f_k$, i.e. a formula in only the variable $k$; and prove the ...
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Tiny Planet Algorithm?

So I've recently been looking at the Tiny Planet images. I've been googling a few things to try and find out how images are converted from normal to a tiny planet. Some phone apps, as well as ...
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Repertoire Method. How to know when equations are valid

I'm trying to work a few examples from the Repertoire Method from the Book Concrete Mathematics. I'm working through the following recurrence: $f(0) = 1$ $f(n) = 2f(n-1) + n$ Which I generalized ...
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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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Strassen's Matrix Multiplication Example Problem

How to multiply two matrices using strassen's matrix multiplication.I have only learned the theory part but i cannot find any examples on the net. Could some one explain with two 2X2 Matrices.
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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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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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Understanding the difference between two similar looking recurrences.

Looking at this recurrence, $f(n) = f(n/2) + nlgn$ The book claims we can conclude that $f(n) = \Omega(nlgn)$ since $f(n/2)>0$. Furthermore, for a sufficiently large $n: lgn \le n^{\epsilon}$ for ...
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Flip all to zero

I have a square grid of size $N$, with rows numbered from $0$ to $N - 1$ starting from the top and columns numbered from $0$ to $N - 1$ starting from the left. A cell $(u, v)$ refers to the cell that ...
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131 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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Recursion, multiplication and exponential

The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} ...
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Finding a recursive definition and computing $B(10)$

For $n \geq 1$, let $B(n)$ be the number of ways to express $n$ as the sum of $1$s and $2$s, taking order into account. Thus $B(4) = 5$ because $4 = 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1 = 2 + 1 + 1 = ...
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Number of ways to arrange n identical stones into piles

Given, there are n stones which are identical. How many ways can the stones be arranged into piles. Suppose if n=4 , we can make one pile with 4 stones or 2 piles with 3,1 or 2,2 or 3 piles with 1,1,2 ...
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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 ...
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88 views

Inductive Definition of regular expression

Give an inductive definition of regular expressions that do not use the star operator. Prove by induction on this definition that every such expression denotes a finite language not containing lambda. ...
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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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60 views

how do you rewrite a recursive formula to find its roots

Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac12 + \sqrt{x_n}$. Rewrite the recursive formula in the form $$ x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c$$ ...
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find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+…+T(\frac n {2^k})$

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+...+T(\frac n {2^k})$ while k is some constant and for any $n\leq3$ $\ T(n)=c$ for k=1 ...
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Discrete Math: “Drop Zero in Bitstring” Recursive function

how do i define “dropzeros” of a bitstring as the result of dropping each 0 from the bitstring. ex) dropzeros of 11 is 11. The dropzeros of 1101 is 111. The dropzeros of 00 is the empty string. Let’s ...
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Discrete Math: Recursive Functions

You are given the following recursive definition defining a set of strings. 1∈S; x∈S → x11∈S. What are the 4 shortest members of the set? What does x11∈S mean?
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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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How to solve recursive function

I've recently been doing some limits with circuits and such, and I came up with the following equation, $R$ being a constant: $$f(x) = \frac{f(x-1)*R}{R+f(x-1)}+R$$ with $f(1)=2$. I know that this ...
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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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Google Question: Number of ways to select sets such that n is pure

Consider a subset $S$ of positive integers. A number in $S$ is considered pure with respect to $S$ if, starting from it, you can continue taking its rank in $S$, and get a number that is also in $S$, ...
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Recurrence Relation for Optimal Card Game Score

I have the following problem where Alice and Bob decide to play a simple card game. At the beginning of the game, $n$ cards are dealt face up in a long row. Each card is worth a different number of ...
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Recurrence relation of an integral [closed]

$$ y_n = \int_0^1 \frac{x^n}{x+5} dx,\ \ n=0,1,2,...,\infty $$ What is the recurrence relation between $y_n, y_{n-1}$ ? Could you provide me a methodology on how to proceed with this?
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Finding the limit of the recursive sequence $r_{n+1} = \sqrt{2 + r_n}$

In the example I am given, I am told that $r_n$ is defined as: $$ r_n = \begin{cases} r_0 = \sqrt{2} \\ r_{n + 1} = \sqrt{2 + r_n} \\ \end{cases} $$ I was told to calculate ...
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Give a randomized algorithm to find the median that has an expected number of comparisons = 2n + o(n)

Any help would be very much appreciated. I'm aware of 2 types of algorithms: "Median of the medians" and one using guards like here: http://www.cs.nthu.edu.tw/~wkhon/random12/lecture/lecture9.pdf ...
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130 views

Solving nested summation $\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}p(A_i)p(B_j) $

I am having trouble solving the following nested summation: $$\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}p(A_i)p(B_j) $$ where $p(A_i) = \frac{1}{n}$, and $n$ is a constant (length of an array). Same goes for ...
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Find another recursive algorithm that is equal to a series

I have the following sequence: $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$ Now I have the following recursive algorithm which is equal to the sequence: $$ y_0 = \log{6} - \log{5} $$ $$ ...
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How to prove that a series is equal to a recursive algorithm

I have the following sequence: $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$ Now I have the following recursive algorithm: $$ y_0 = \log{6} - \log{5} $$ $$ y_n = \frac{1}{n} - 5y_{n-1}, n ...
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How does my textbook solve this summation equation for the answer?

Summations have always been my weakness in mathematics, and it's showing here as I'm very confused how my textbook, Introduction to Algorithms, goes from basically the second half of the following ...
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What is the bound of: $T(n) = T(n-2) + (n)log(n)$?

I am given the following recurrence relationship: $\ T(n) = T(n-2) + nlog(n)\\ T(1) = T(0) = constant$, I need to find the order for the recurrence. So, using the iterative methodology, what I ...
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How does my textbook come up with this statement? I don't believe it to be true.

My textbook (Introduction to Algorithms) states the following: When polynomially comparing $n^\epsilon$ and $lgn$, it states that $n^\epsilon$ is polynomially greater for any positive $\epsilon$. ...
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Where is the value of the variable $\epsilon$ obtained in the following explanation my professor gave?

My professor gave us this explanation from the textbook Introduction to Algorithms regarding the Master Method/Theorem: As a first example, consider $$T(n)=9T(n/3)+n.$$ For this recurrence, we ...
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238 views

Big Oh Notation for a Recursive Algorithm

I have a question that I'm unsure of: Express the complexity of the following method using big-O notation. You must explain how you arrived at your answer. What value is returned by the call ...
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An alternative to the Master Theorem?

I've been reading up on the Master Theorem (used for identifying the worst case run times of a certain class of recursive functions [ the divide and conquer type] ). I looked at the Wikipedia entry ...
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Termination of a Fast Exponentiation problem

Here's the problem I am stuck on. There exists a fast exponentiation program like the following: Given inputs a in the set of all Real numbers, b in the set of Natural numbers, initialize ...
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How does my professor come up with the recursive case in this algorithm analysis?

My professor gave us the following explanation for the recursive algorithm for finding the permutations of a set of numbers: When he has (T(m+1), n-1)) where does that come from? Why is it m+1 ...