Questions tagged [recursive-algorithms]
Questions dealing with recursive algorithms. Their analysis often involves recurrence relations, which have their own tag.
1,376 questions
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How to make a recursive function that converges to a specific value after a certain amount of time [closed]
I am making a video game, and I am currently trying to get the character's acceleration feeling good. I have come up with a basic function that will determine the character's speed after a certain ...
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Minimum steps to cut a truss
Motivation
While I was looking at the method of sections to find the forces in the members of a truss in engineering mechanics, I had a question about the minimum amount of calculation required to ...
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Unorthodox implementation for a recursive identification method
Recently I realized that I've coded a custom-made function for recursive output error method which differs a bit from the traditional algorithm and would like to know the perception of my peers. ...
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Conjectured fast and simple recursive algorithm for Bernoulli numbers
Let
$B_n$ be the $n$-th Bernoulli number.
$T(n,k)$ be an integer coefficients such that $T(n,k) = \nu_k$ where we start with vector $\nu$ of a fixed length $n$ with elements $\nu_1=1$, $\nu_i=0$ for $...
user929945
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Fast and simple recursive algorithm for A375540
Let
$a(n)$ be A375540, i.e., an integer sequence such that $$ a(n) = 2^n n! [x^n] (1/2 - \exp(-x))^n. $$
Start with vector $\nu$ of fixed length $m$ with elements $\nu_i = 1$ (that is, $\nu = \{1,1,\...
user929945
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If $f$ is an autosimilar function s.t. $f(x)=\sum_{l\in A}p_l f(c_l x)$, then $f(x)=\sum_{w\in W}p_wf(c_w x)$ for cylindrical $W\subset A^*$
I think that I have a proof for the following claim, but I am looking both for a feedback and suggestions how to make it better, for example how to make it a top-down rather than a bottom-up type of ...
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How to informally understand `hash reducibility`?
I was reading a paper on a certain reducibilities in Polynomial classes. And I stumbled across this definition: For sets $A,B$ computable in polynomial time, we write $A \leq^{\#} B$ via a polynomial $...
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Trouble understanding quicksort O(nlogn) proof
I have trouble understanding this proof for quicksort: The goal is to bound the overall number of comparisons performed by the algorithm. Consider an array $A$ with elements $z_1, z_2, ..., z_n$ with $...
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Is my proof of $T(n) = T(n - 1) + n$ being $O(n^2)$ correct?
I know that this has been posted at least once, but the accepted solution doesn't use the substitution method laid out in Introduction to Algorithms by Thomas Cormen.
What I've tried so far is this:
$...
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The growth rate of a recursively defined function
Imagine a function where you know how to get from $f(n)$ to $f(n-1)$ and from $f(2n)$ to $f(n)$. How many steps would it take to get to $f(1)$? I am not talking about the optimal route but rather if $...
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Finding order of time complexity for program finding the $m$-subsets of an $n$-set.
For the code here, the analysis for the order-of-time complexity is as follows:
For the purpose of finding the time-complexity of the above program; the program statements of concern are:
...
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Redistributing points along a set of points (Stroke)
I'm currently facing the following problem: I want to write an algorithm that, given as input a list of points (representing a path/stroke/curve), and a target $distance$ it creates a new stroke that ...
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Minimum distance decoding
I have been researching decoding methods for Reed–Muller codes and am currently reviewing a recursive decoding approach proposed by Ilya Dumer in his paper “Recursive Decoding and Its Performance for ...
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Algorithm to expand arbitrary partitions
Suppose you have an expression like:
$$
(a + b + c + d)(e + f + g)(h + i)
$$
Is there a method to expand this expression that does not involve multiplying only two groups at a time? Some way to ...
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Recursive to explicit/closed form [duplicate]
A similar question has been asked before but I haven't seen an answer to this specific one after looking for so long.
I have a recursive function where:
$P(2) = 1$
For $n ≥ 3$, $P(n) = (n-2)P(n-1) + ...
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Solve Recurrence Relation with Master theorem Case 3
I need your help with the following problem:
I’m supposed to find a simple function $ g(n) $ for the recurrence relation
$ T(n) = 9 \cdot T\left( \lceil \frac{n}{4} \rceil \right) + 3^n $
with the ...
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Recursive system identification initialization (continuous-time model)
I have a doubt concerning recursive system identification. I have seen that in many occasions we have a data set $u(t)=[u(1)\, u(2) \, \dots]$. When the first couple of data $y(t)$ and $u(t)$ arrives,...
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Asymptotic equivalent of sequence defined by a recursion relation.
For the study of a termination speed of a recursive algorithm of mine, I would like to have more precise result on the following sequence:
$$0< a_0 = a< 4 \qquad \text{and} \qquad a_{k+1} = \...
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Explain how to find this formula from the recursion in this paper
I am trying to understand how to find this formula, which seems straight forward, but I am missing something.
In page 20 of this paper (or thesis):
https://egrove.olemiss.edu/cgi/viewcontent.cgi?...
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Strict bound for recurrence formula
Given the recurrence $$T(n) = (2k - 1)T(\frac{n}{k})+2^kn$$ I need to show (or disprove) that for every $\epsilon>0$ there exists a $k$ such that $T(n)=o(n^{1+\epsilon})$.
So far, I've been able to ...
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solve a recursion formula using a generating function
I have the following formula for a vector $v_n= \ ^t(a_n, b_n)\in \mathbb R^2$ :
$v_n= Av_{n-1}+ Bv_{n-2}$ for two given matrices $A$ and $B$ in $M_2(\mathbb R)$, and we are given $v_1$ and $v_2$ of ...
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Recursive sum of binomial random variables
In my work I've come across the following problem.
For $Y_k\sim\text{Binomial}(X_k,p)$ and $X_0=1$, compute the probability distribution for the following recursion,
$$X_k=X_{k-1}+Y_{k-1}$$
This ...
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Solution to recurrence equation $T(x,y) = T(x-1,y) + T(x,y-1) + O(1)$ in terms of big-O expression
What would be the solution, expressed in big-O notation, to the recurrence equation $$T(x,y) = T(x-1,y) + T(x,y-1) + O(1)$$ with base case being either or both $$T(x,0) \text{ for any x}$$ $$T(0,y) \...
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Prove or give a counterexample of a recursively defined function
Conjecture:
For any given $\alpha\in\mathbb{R},$
Any given bijective continuous function $f:D \rightarrow \mathbb{R_{\ge\alpha}},$
And any given $\mu_\alpha:D\rightarrow \mathbb{R}$, where $D\subset\...
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I stumbled on a bruteforcing median algorithm. Can anyone tell me why it works and is there a name for this equation?
I recently stumbled on an algorithm while trying to write a program to estimate a median without sorting, a randomized group of non-repeating numbers with a known number of elements. Performance is ...
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How to calculate $p_{A}+ p_{B}$ quickly?
Given $p_{A}, p_{B}, p_{C}$ so that $p_{A}+ p_{B}+ p_{C}= 1$, and the transition matrix equation $\begin{bmatrix} a+ m & -b & -s\\ -a & b+ d & 0\\ -m & -d & s\\ \end{bmatrix}\...
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Prefix-free binary codes with costs
I came across the following problem:
We would like to generate a prefix-free binary code with $n$ codewords, such that we pay $4$ units for each 1-bit and $1$ unit ...
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resolve recurrence $a_n=a_{n-1}+3a_{n-3}$
Hy all
I want to try to find the solution for the recurrence $a_n=a_{n-1}+3a_{n-3}$. If i take the equation associate to it, then i have $x^3-x^2+3=0$. But in this case, i have just one real root and ...
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Find the chromatic polynomial of a graph using deletion and contraction
Find the chromatic polynomial of the following graph.
The graph in question is the one on top and I am doing deletion and contraction recursively as the picture shows.
This is what I get and I have ...
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In how many ways can the integers $1,2, \dots ,n$ be arranged so that there is only one integer that is immediately followed by a smaller integer? [closed]
I attempted this using an recursive idea. I noticed that as long as $1$ does not appear on the first position and it appears on the $k$-th position, then there are exactly $n$ choose $k$ number of ...
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$\mu$ recursive function exercises
Could you please explain the solutions of the exercises? The definition of $\mu$- recursive finctions is clear to me, but I want to know the way of thinking to solve the exercises.
Thank you.
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Proving this recursive merging routine from merge sort works
I was wondering if anyone could provide a proof that the following algorithm to merge $2$ sorted arrays into a combined sorted array works:
\begin{array}{l}
\texttt{function merge($x[1 \ldots k], y[1 \...
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Dynamic programming, optimization problem where decision leads to multiple overlapping subproblem. [closed]
In the matrix chain multiplication (MCM) problem each time we apply a decision of parentizing an expresion $e=(e_1)(e_2)$ we have two subproblems to solve but they are not overlapping. Indeed, solving ...
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Reasoning about the Collatz conjecture, multiple infinitely growing trees that never overlap? [closed]
I have been pondering the Collatz conjecture recently as a mental exercise, and have run into a problem that has to do with proving that an iteratively growing tree of odd positive integers will ...
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Generate set of numbers containing 3 consecutive 1, but without the elements of the previous set [closed]
So I have this specific problem that I couldn't figure out. I want to create a set $F_n$ containing all bitstrings that has 3 consecutive 1s, but not those that are already contained in all the ...
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Division based recurrences instead of subtraction based: $F(x)=F(x/2)+F(x/3)$
The most famous (and simplest non-trivial) recurrence is the Fibonacci recurrence $F(n)=F(n-1)+F(n-2)$ with $F(0)=0, F(1)=1$. What if we consider instead division based recurrences, the simplest non-...
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Why doesn't this diagonal argument work?
I have a question about the standard rules for computing p.r. terms (see below). It seems pretty clear that these rules could be used to define a p.r. operation that evaluates any p.r. term of the ...
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How to Prove Division of One Bezier Curve into Two using de Casteljau Algorithm
$\newcommand{\brstbasis}[2]{b^{#1}_{#2}}$
$\newcommand{\posintset}{\mathbb{Z}^{+}}$
$\newcommand{\intset}{\mathbb{Z}}$
$\newcommand{\realset}{\mathbb{R}}$
$\newcommand{\domain}[1]{\operatorname{dom}\...
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Recursive approximations of inverse square law
I have a toy electrostatics simulation that consists of some number of 2D point particles that each have a real-valued "charge" $q_i$, which then exert forces on each other proportional to $...
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Tried finding an efficient algorithm for a 4-digit number guessing game, knowing only the number of digits on correct positions..
I've been playing a game similar to Bulls and Cows, but it goes like this: one player has to pick a random $4$ digit number. Digits can repeat, any digit between $0$ to $9$ and, you only get the ...
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Find limit of Decrementing Recursive Series
I want to find a formula to find the lower limit part of this recursive or geometric series
$$
x_{n} = \left( x_{n-1} + p \right) \times \left( 1 - \frac{t}{100} \right)
$$
I was just wondering what ...
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Prove that $x_{n+1} = \frac{1}{3}(2x_n + \frac{a}{x_n^2})$ is decreasing
Prove that $x_{n+1} = \frac{1}{3}(2x_n + \frac{a}{x_n^2})$ is decreasing where $x_1$ $> 0$.
I have been asked the above question and the working out given to me skipped some steps in between.
It ...
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create a recurrence relation for the number of ways of creating an n-length sequence with a, b, and c where "cab" is only at the beginning
This is similar to a problem called forbidden sequence where you must find a recurrence relation for the number of ways of creating an n-length sequence using 0, 1, and 2 without the occurrence of the ...
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Lemma 6.2. in Scaling Algorithms for the Shortest Path Problem
I have a question regarding the proof of Lemma 6.2. in this paper: https://www.cs.princeton.edu/courses/archive/fall03/cs528/handouts/scaling%20algorithm%20for%20the%20shortest.pdf.
The simplified ...
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Let $x_0 = 3;\ x_{n+1}=3x_n\ $ if $\ \frac{x_n}{2}<1;\ x_{n+1}=\frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\ $ Is $\ \liminf_{n\to\infty} x_n=1?$
This is a natural follow-up question of this previous question of mine.
Let $x_0 = 3.$ Let $\ x_{n+1} = 3x_n\ $ if $\ \frac{x_n}{2}<1;\quad x_{n+1} = \frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\quad ...
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Guaranteed graph labyrinth solving sequence
Starting from a vertex of an unknown, finite, strongly connected directed graph, we want to 'get out' (reach the vertex of the labyrinth called 'end'). Each vertex has two exits (edge which goes from ...
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Finding an algorithm that goes through all the possible permutations of a set only by swapping 2 elements
I recently came across a problem when trying to deal with a set of numbers with n elements.
The problem is as follows:
Starting with a set of n distinct elements, how would one generalize a unique ...
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Subset Product has a pseudo-polynomial algorithm?
Subset Product- Given $N$ and a list of positive divisors $S$, decide if there is a product combination equal to $N$
We will sort $S$ in numerical order from smallest to largest in order
to find out ...
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Prove or reject my solution to "How quickly can you type this unary string?"
I had posted an answer on the Code Golf SE yesterday. Although the answer on that site remain valid if no counterexample can be find. I'm interesting in its correctness. So I want to find a prove or ...
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What is the set generate by ${1}$ and a function $1/(a+b)$?
If I'm given a starting set, and an operation, what would the generated set looks like?
Here we take $S_0=\{1\}$ and $f(a,b) = \dfrac{1}{a+b}$ as an example, the following Mathematica codes shows the ...
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