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

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3
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0answers
123 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 ...
4
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0answers
106 views

How can I convert a recursive equation to a non-recursion one? [duplicate]

While I am aware of several other similar question which have been asked in the past, my mathematical education only extends through calculus, making them beyond my comprehension, and I believe this ...
5
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2answers
424 views

Characteristic equation. What is it?

Consider a recursive equation: $$ a_n = A a_ {n-1} + Ba_ {n-2} $$ And for this type of equation, there is characteristic equation: $ x ^ 2 = Ax + B $. What does exactly means that this equation is ...
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0answers
46 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 ...
1
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1answer
345 views

Give tight asymptotic bounds for the following recurrence. T(n) = 3T(n/3)+log n

Give tight asymptotic bounds for the following recurrence. Justify your answers by working out the details or by appealing to a case of the master theorem $$T(n) = 3T\left(\frac n3\right)+\log(n)$$ ...
4
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1answer
43 views

Finding all local $k$-maximums in sequence $a_1, a_2, \ldots a_n$.

For given sequence of numbers $a_1, a_2, \ldots, a_n$ we say that $a_i$ is $k$-local maximum, if $i > k$ and $a_i$ is largest of numbers $a_{i - k}, a_{i-k+1}, \ldots, a_i$. How can we find all ...
3
votes
1answer
68 views

Finding $m$ largest numbers from union of $k$ sorted lists $A_1, A_2, \ldots, A_k$

We are given $k$ sorted lists $A_1, A_2, \ldots A_k$ with corresponding sizes $n_1, n_2, \ldots n_k$. How can one find $m$ largest elements (numbers) from union of lists $A_1, A_2, \ldots, A_k$? We ...
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0answers
100 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 ...
3
votes
1answer
45 views

Question about recursive algorithm

I have following problem: $$f(n)=\frac{1}{1^2+1}+\frac{2}{2^2+1}+\frac{3}{3^2+1}+\cdots+\frac{n}{n^2+1}$$ Write recursive algorithm for $f(n)$ Prove that recursive algorithm is correct ...
3
votes
1answer
126 views

How can I recursively approximate a moving average and standard deviation?

Consider a sequence of measurements $(x_1, x_2, ...)$. Let $\mu_n$ be the $p$-period moving average defined by $$\mu_n = \frac{1}{p}\sum_{i=n-p+1}^nx_i$$ and $\sigma_n$ be the $p$-period moving ...
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2answers
365 views

Dynamic programming:Making a Change

I'm practicing problems on dynamic programming.The problem is as follows: You are given n types of coin denominations of values v(1) < v(2) < ... < v(n) (all integers). Assume v(1) = 1, so ...
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1answer
27 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
0
votes
1answer
42 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
0
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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
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4answers
353 views

Prove that the Fibonacci recursion diverges

I have this sequence with $ n \in \mathbb{N} $ $ f(1) = f(2) = 1 $ and $ f(n) = f(n-1) + f(n-2) $ for $n \ge 3$ I think this sequence is bounded below and unbounded above. So it's clear that this ...
0
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3answers
66 views

Constructing a recursive sequence that converges to sqrt 17

One of the problems that we have for abstract math is the following: Using the recursive sequence definition, construct a sequence that converges to $\sqrt{17}$. It is my understanding that the ...
4
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1answer
145 views

All solutions of the recurrence relation

Find all solutions of the recurrence relation $$ a_n = 2a_{n-1}+15a_{n-219}-64a_{n-3}+k $$
0
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1answer
47 views

Explicit (General) formula for recursive definition.

I am given $a_n=3a_{n-1}+4^n$, $n=1,2,3,....$ and $a_0=1$. First four terms: $$ \begin{align} a_1&=3.1+4^1=3+4=7\\ a_2&=3.7 + 4^2 = 21 + 16 = 37 \\ a_3&=3.37 + 4^3 = 111 + 64 = 175\\ ...
0
votes
1answer
47 views

Solving time complexity of merge sort

I was asked to prove that the time complexity of merge sort is $ O(log_2n)$ but I cannot find a way to continue my method. Any help? $T(n)=2T(\frac{n}{2} )+n$ $T(n)= 2[2T(\frac{n}{4})+n] +n = ...
0
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1answer
31 views

Log property in proof of Master theorem

The family of recurrence considered is of the form $$ T(n) = aT(n/b) + n^c $$ $a,b,c$ are integers. One case of master theorem states: if $c< log_b a$, then $T(n) = \Theta(n^{log_b a}) $. I have ...
4
votes
2answers
140 views

Finding the convergent value of a recursion similar to Arithmetic-Geometric Mean recursion

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
2
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4answers
99 views

Find exact closed form of recurrence $g(0) = 0, g(1) = 3, g(n) = g(n − 1) + 2g(n − 2)$ for $ n \geq 2$

$g(0) = 0, g(1) = 3, g(n) = g(n − 1) + 2g(n − 2)$ for $n \geq 2$ Our lecture notes suggest us to work backwards until you get the first term, i.e. $g(1)$ I am not quite sure how that works as the ...
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0answers
61 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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0answers
157 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} ...
0
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1answer
68 views

A Difficult Recursive Equation

I've got a recursive equation of the form $$ x_{n+1} - x_{n} = \frac{(-1)^n}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}(x_0-x_1)$$ for $n \geq 2$. We can assume $x_0$ and $x_1$ are just real numbers/ I ...
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3answers
25 views

running time of an algorithm

I am trying to prove an algorithm with input size $n$ satisties the recurence relation (for $n>=1$) $T(n) = T(n-1)+n$ and an initial condition of $T(1)=1$ has running time in $Θ(n^2$). By using ...
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0answers
31 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 ...
0
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1answer
59 views

$T(n)=T(n-1)+O(\log n)$ is $T(n)=O(n^2)$ or $T(n)=O(n \log n)$

I have this Recurrence relation: $T(n)=T(n-1)+O(\log n)$ What is the solution? $T(n)=O(n^2)$ or $T(n)=O(n \log n)$ What I did is: I assume that $T(n)\le O(n^2)$ And that's bring me to $O(n^2)$, ...
2
votes
2answers
159 views

Towers of Hanoi Starting From Initial (Legal) Configuration?

I was recently asked in an interview how an algorithm for solving the classic Towers of Hanoi problem would differ if you were given an initial (legal) configuration of the towers, and had to start ...
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3answers
67 views

recursive to explicit sequence

I am trying to find the explicit formula for the following recursion: $$a_{1}=3,\quad a_{n}=3- \frac{1}{a_{n-1}},\quad n \in \mathbb N,n>1$$ I tried in many ways but I cannot find any ...
1
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1answer
28 views

Algorithm Running Time

Given an algorithm with a running time $$T(n)=5T(n/2)+n^2$$ So the number of nodes at a depth $i$ would be: $5^i$ The input size for each node at $i$ would be: $n/2^i$ Agreed. Then it states that ...
5
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1answer
529 views

How many distinct ways to climb stairs in 1 or 2 steps at a time? (Fibonacci puzzle)

There is a very interesting puzzle for fibonacci sequence ...
0
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1answer
38 views

How to find if an array has at least 10 unique integers in $O(\log n)$?

I am given a sorted array of integers. I want to find out if the array has at least 10 unique integers. I know this can easily be done with an algorithm that runs in $O(n)$ simply by going through ...
1
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0answers
41 views

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 ...
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 ...
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2answers
53 views

Optimal Number of White Balls

There are C containers, B black balls and infinite number of white balls. Each container should have at least one ball. Each of the containers may contain any number of black and white balls. Action ...
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0answers
47 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}\{ ...
1
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1answer
359 views

Generate all permutations of a string containing repeated characters

I was writing a program to print all the permutations of a string. I came up with the following: ...
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0answers
25 views

Binary Search when no “fixed” answer…? (Font Sizing)

So I'm trying to figure out how best to implement a binary search algorithm to find the "optimal" font size for a piece of text to fit into a given space, to the nearest 0.5pt. My understanding is ...
0
votes
1answer
22 views

Recurence Problem. - Solve either by substitution or Expansion

Function T(n) is defined by the following recurrence relation: $$ T(n)=2T(\lfloor\sqrt{ n}\rfloor)+\log(n) $$ $$ T(0)=1 $$ How would I Solve by substitution and/or Expansion? Note: ...
1
vote
1answer
109 views

Using Extended Euclidean Algorithm for $85$ and $45$

Apply the Extended Euclidean Algorithm of back-substitution to find the value of $\gcd(85, 45)$ and to express $\gcd(85, 45)$ in the form $85x + 45y$ for a pair of integers $x$ and $y$. I have ...
0
votes
2answers
57 views

A nice recurrence sequence

I want to find general solution for recurrence: $a_n=7a_{n-1}-10a_{n-2}+4n$ with suppose $c_1, c_2, c_3$ is constant. I need some tutorial or solution on this challenging recurrence.
1
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1answer
321 views

Solving recurrence equations with repeated substitution?

Say we have a recurrence equation as $$ T(n) = \begin{cases} T\left(\frac n2\right) +n & \text{if }n\ge2 \\ 1 & \text{if }n=1 \end{cases} $$ Would the first substitution be like this? ...
4
votes
1answer
61 views

Solving recurrence -varying coefficient

How can one find a closed form for the following recurrence? $$r_n=a\cdot r_{n-1}+b\cdot (n-1)\cdot r_{n-2}\tag 1$$ (where $a,b,A_0,A_1$ are constants and $r_0=A_0,r_1=A_1$) If the $(n-1)$ was not ...
0
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1answer
177 views

Karatsuba multiplication algorithm of two 6 digit decimal numbers.

What are the min and max number of single digit multiplications, involved in recursive karatsubha multiplication of two 6 digit decimal numbers? I found different results on different numbers. But I ...
0
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1answer
131 views

Improving the Edit Distance Algorithm

I applied an Edit Distance Algorithm for similarity between two strings over the lowercase latin alphabet, where the first string has length $m$ and the second length $n$. However I want to improve ...
0
votes
1answer
121 views

Any way to get a Recursive function from its closed form?

For an exercise, the goal is to find a recursive definition for a certain function. the function itself is as follows: f(a,b) is the # of binary strings of length a and with b more 1s than 0s. eg: ...
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1answer
31 views

A recurrence equation for the number of multiplications in an algorithm for computing powers

We were asked the to find the recurrence equation for the number of multiplications for the following algorithm as a function of N. For simiplicity, let $ N = 2^k $ and is a positive integer: I ...
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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
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1answer
53 views

Geometric sum of recurrence relation

I am reading the textbook Algorithm Design by Kleignberg and Tardos and I am having trouble on page 216. $$T(n) \le \sum_{j=0}^{log_2n - 1} \left ( \frac{q}{2}\right )^j cn = cn \sum_{j=0}^{log_2n - ...