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

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61 views

Greek cross fractal

I need some code to generate a Greek cross fractal. Example: It must be made of increasingly smaller panels, but the panels may not overlap with previous panels. Does anyone know where I might ...
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3answers
115 views

Solving the recurrence relation $T(n) = 2T(n^\frac{1}{2}) + c$

I've been trying to do this for hours. I just don't know how. I'm familiar with recurrence relations in the form of $T(\frac{n}{2})$, but what do you need to do to solve $T(n^\frac{1}{2})$? I've ...
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0answers
13 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
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1answer
109 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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25 views

recursion $t(n)=\sqrt{2} \times \frac{tn}{2} +\log{n}$

I tried substituting $m=\log{n}$ $t(2^n)=\sqrt{2} \times \frac{t2^n}{2}+m t(m)= \sqrt(2) \times \frac{tm}{2}+m$ From here I got $\log {n}$ But with induction I proofed its $\sqrt {n}$
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1answer
933 views

Finding Pareto optimal solution set in $O(n \log n)$ time

http://cs-people.bu.edu/kvodski/teaching/spring10/lab7.html says: For two points in 2-dimensional space, point ($x_i$, $y_i$) dominates ($x_j$, $y_j$) if $x_i > x_j$ and $y_i > y_j$. Given a ...
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1answer
33 views

Is there an algorithm to generate these specific sequences of numbers?

f(1) = [1] f(2) = [2,1,1] f(3) = [3,2,1,1,2,1,1] f(4) = [4,3,2,1,1,2,1,1,3,2,1,1,2,1,1] ... f(n) = ... The lengths of the lists f(n) are $2^n - 1$ (Mersenne ...
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1answer
43 views

Why $T(n) = 2T(n/2) + \log n$ is in $ O(n)$?

My professor said that $T(n) = 2T(n/2) + \log n$ is in $O(n)$ I checked with Master Theorem and I did not really understood why. By Case1 (which would give us exactly $O(n)$) we have $a = 2, b = 2, ...
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29 views

Values of the Sudan function

I am talking about the first discovered recursive function which is not primitive recursive. I would like to know the exact values of $\ f(3,3,3), f(2,0,4), f(2,7,1), f(2,3,2)$ (where $f$ is the ...
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1answer
355 views

Recursive Function for Pyramid-Scheme

consider a group which has 1 user. each month, every user can bring another user to join the group. the user that has been joined for 3 months, should leave the group. calculate the total membership ...
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0answers
20 views

FFT procedure for evaluationg a polynomial at $N$ Fourier points

The following is the recursive FFT procedure of Algorithm for evaluationg a polynomial of length $N$ at $N$ Fourier points. Algorithm (FFT - fast Fourier transform). Input arguments. $ \ ...
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1answer
76 views

How can I solve this recurrence problem?

Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
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1answer
26 views

Show that running time of Quick Sort is $\mathcal O(n^2)$ when array contains distinct elements and is sorted in descending order

I'm trying to study running time for various algorithms. Now I have QuickSort. How exactly is the running time of an algorithm calculated, I know how quick sort works and the Asymptontic notations. I ...
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1answer
15 views

Understanding recursive function for finding GCF of 2 numbers

So I get how this code works, but I don't understanding why it works. The function assumes input num1 > num2. Algorithms are hard for me to grasp, so please explain to me like I'm five. Heres the ...
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1answer
340 views

How to find a closed form formula for the following recurrence relation?

I have to find a closed form formula for the following recurrence relation which describes Strassen's matrix multiplication algorithm - $$T(n) = 7\,T\left(n \over 2\right) + \frac{18}{16}n^2$$ with ...
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0answers
10 views

Bounded Knapsackproblem Formula DP

I knew how the binary Knapsack works with Dynamic Programming. But, now I am interested. How does the recursive formula look like if I allow n€{0,1,2} of the same item to be in the Knapsack? The only ...
2
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2answers
463 views

how to show the convergence of an algorithm

I have two unknown variables x and y which are non linear equations to be solved. \begin{eqnarray} y=\frac {|\sin(2x+\theta)|}{\sin x\sqrt{A+2B\cos(2x+\theta)}} \nonumber \\ x=\arccos\bigg( ...
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6answers
777 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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35 views

Solving this recurrence relation representing constant-power loads on a resistive cable

Given the following: $$\begin{align} v_n&=v_{n-1}-r\sum_{i=n}^m \frac p{v_i}\\ v_0&=V \end{align}$$ where: $$\begin{align} m\ge n\ge 0\;&:\;m,\,n\in\mathbb {N_0}\\ ...
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0answers
16 views

Sources and sinks for parabolic PDE algorithm

I am given a very basic fortran program (View here) and asked 1st to investigate its accuracy and stability, for various values of $ \Delta t $ and lattice spacings. The program is an implementation ...
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3answers
4k views

What will the recursion tree of Fibonacci series look like?

I am watching the Introduction to algorithm video, and the professor talks about finding a Fibonacci number in $\Theta(n)$ time at point 23.30 mins in the video. How is it $\Theta(n)$ time? Which ...
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1answer
24 views

Find the asymptotic tight bound for $T(n)=T(n-1)+n lg n + n$ and for $T(n)=n^2 \sqrt{n}T(\sqrt{n})+n^5lg^3n+lg^5n$

I am stucked at this problem: Find the asymtotic tight bound for the following recurrences: (Assume that $T(n)$ is constant for sufficiently small $n$) (1) $T(n)=T(n-1)+n lg n + n$ (2) $T(n)=n^2 ...
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3answers
32 views

A set S is defined recursively by

A set S is defined recursively by Basis step: $0 \in S$ Recursive step: if $a \in S$, then $a + 3 \in S$ and $a + 5 \in S$. Determine the set $S \cap \{ a \in \mathbb Z \mid 0 < a < 12 \}$. ...
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0answers
58 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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2answers
49 views

Probability of arriving H before A [closed]

I m in the point X. I m 2 blocks up from a point A and 3 blocks down from my home H. Every time I walk one block i drop a coin. H . . . X . . A If the coin is face I go one block up and if it is ...
2
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2answers
62 views

Most efficient algorithm to distribute n n-bit strings among n people

We are given $n$ people, whom we identify with the elements of $[n]=\{1,\ldots,n\}$. We are also given a finite collection $\mathcal{K}$ of subsets of $[n]$. The problem is to (efficiently) ...
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58 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
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4answers
881 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 ...
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1answer
149 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 ...
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3answers
239 views

Calculating a SQRT digit-by-digit?

I need to calculate the SQRT of $x$ to $y$ decimal places. I'm dealing with $128$-bit precision, giving a limit of $28$ decimal places. Obviously, if $\,y > 28$, the Babylonian method, which I'm ...
2
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1answer
41 views

Is this the correct minimum number of coins needed to make change?

The Problem: On Venus, the Venusians use coins of these values [1, 6, 10, 19]. Use an algorithm to compute the minimum number of coins needed to make change for 42 on Venus. State which coins are used ...
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1answer
25 views

How would you apply the Greedy technique in this situation/why wouldn't it work?

I am going over the Rod Cutting Problem The author states "Selling a rod of length $i$ units earns $P$[i] dollars." Here is the table $P$ for this problem I'am currently going over this question ...
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1answer
33 views

Where does 13 come from?

I am going over the Rod Cutting Problem Everything makes sense to me until For example, $L$ = {9} has the total cost Cost($L$) = $P$[9] = 13, whereas $L$' = {1,1,1,1,1,1,1,1,1} has the total cost ...
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1answer
37 views

Sequence who converges to $\sqrt{a}$ for every $a\geq0$

If we have: $$x_n=\left\lbrace \begin{matrix} b\in\mathbb{R}\setminus \{0\} & ,n=1 \\ \dfrac{a+x_{n-1}^2}{2x_{n-1}} & , n>1\end{matrix} \right.$$ Then, is easy to prove that $(x_n)\to ...
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1answer
63 views

Solving $T(n) = 3T(n-1)$

How is the constant before the $T$ important to the result from $T(n)$ I know that \begin{equation*} T(n) = T(n-1) + 3 \Rightarrow \theta(n)~\text{and}~T(n) = T(n-1) + n \Rightarrow \theta(n^2) ...
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1answer
88 views

Struggling with difference between greedy and naive but optimal algorithms? (Graph theory)

I've been thinking about the following problem for quite a while and tried multiple solutions, but I'm having difficulty telling the difference between a greedy algorithm and an inefficient naive ...
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1answer
29 views

Where does the root of this tree come from?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
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3answers
46 views

How to apply Master theorem to this relation?

This is the definition of master theorem I am using(from Master Theorem) I am trying to use that master theorem to find the tight bound for this relation $T(n) = 9T(\frac{n}{3}) + n^3*log_2(n)$ ...
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1answer
46 views

Wouldn't this Greedy Algorithm achieve the highest possible of money in this situation?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
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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 ...
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0answers
12 views

recurrence tree final step - binary search

Starting with the base case and recursive case run times as follows: 􏰀 t(N) = 1 , if N = 1 t(N)= 1+t(N/2) ,ifN > 1 At the end of my tree I have ...
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0answers
12 views

Formula for a recursive function

Given the recursive function $T: \mathbb{N}_0 \to \mathbb{R}_+$ $$T(n) ≤ max_{1 ≤ k ≤ n - 1}\{T(k-1) + T(n - k) + c n^2\}$$ with c > 0 constant, I want to determine an absolute formula to quickly ...
0
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1answer
24 views

Recursive solution to “Number of subsets of a set without 3 following numbers”

so I got the following problem: For a given natural n number, let A be the set A={1,2,...,n} I need to provide a recursive solution that will return the number of possible subsets that doesn't ...
0
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1answer
23 views

Why is $X_m$ and $Y_m$ not included in the shaded region(where median can lie)?

This problem is from Algorithms, problem 2 The Problem Given two sorted list of numbers $X$[1..$n$] and $Y$[1..n]. we need to come up with a O($\log n$) time algorithm to find the median of the 2$n$ ...
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2answers
26 views

Getting rid of $2^n$ when solving $a_n=8a_{n-1}-20a_{n-2}+16a_{n-3}+2^n$ by characteristic roots

$a_n=8a_{n-1}-20a_{n-2}+16a_{n-3}+2^n$ For $n\ge3$, With initial conditions $a_2=1$, $a_1=1$, and $a_0=1$ I'd like the find the particular solution with characteristic roots. However when generating ...
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0answers
58 views

Closest pair points algorithm with Manhattan Distance

According to the solution of closet pair (euclid distance) with divide and conquer algorithm during merge algorithm we prove that for each point at distance d (minimum distance of two different sub ...
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0answers
20 views

How to state a recurrence that expresses the worst case for good pivots?

The Problem Consider the randomized quicksort algorithm which has expected worst case running time of $\theta(nlogn)$ . With probability $\frac12$ the pivot selected will be between $\frac{n}{4}$ and ...
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2answers
25 views

Does this recurrence relation run in $ \Theta(n) $?

This is the recurrence relation I am trying to solve: \begin{align} T(n) & = 2 \cdot T \left( \frac{n}{4} \right) + 16, \\ T(1) & = c. \end{align} I broke this down (i.e., solved this ...
2
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1answer
42 views

How to show that recurrence $T(n) \in \Omega(n^{0.5})$ using proof by induction?

This is recurrence $T(n)$ $ T(n) = \begin{cases} c, & \text{if $n$ is 1} \\ 2T(\lfloor(n/4)\rfloor) + 16, & \text{if $n$ is > 1} \end{cases}$ This is my attempt to show that $T(n) \in ...
2
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0answers
88 views

How to find a enclosed envelope with maximum points among a cloud of spheres? [ Concave Hull ]

I have a questions regarding the selection of the outer points (i.e. sphere center), among a collection of many spheres in 2D/3D space. The outer points is that when all of these points are ...