Computational complexity, a part of theoretical computer science that deals with understanding how efficiently a problem can be solved.

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Factoring semiprimes cost estimation

I have two problems that are the following. The first problem is the following: I need to estimate the cost of factorizing a given semiprime based on previous estimations. For example I have the time ...
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Algorithmic complexity from knowing complexity due to two different factors.

I have an algorithm that has complexity depending on two factors $n$ and $m$. If I know that fixing $m$ I have complexity $\mathcal{O}(n^p)$ and fixing $n$ I have complexity $\mathcal{O}(m^q)$, can I ...
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Omitting the refference to a particular logarithmic base - order notation

How can I prove by using the Order notation definition that we can conventionally refer to an algorithm taking "log time", without referring to a particular logarithmic base?
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Solving the Gobblet game

In 1995 the Connect-4 Game was solved with a brute force approach. Using the standard 6 high / 7 wide grid, first player can force a win in 41 moves. Complexity of the Connect-4 game could be ...
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What will happen if any language in NP ∩ co-NP will become NP-complete?

I approached this question like this: Let B ∈ NP ∩ co-NP and B is also NP-complete. Then any other problem in NP can be reduced to B. Now take A ∈ co-NP. Then ~A ∈ NP which can be reduced in ...
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Is $O(x^2)$ equal to OR a tighter bound for $O(x(x-y))$ if $x, y >0$ and $x>y$ alway hold?

In the question, $O$ is the Big-O notation, please see https://en.wikipedia.org/wiki/Big_O_notation. $x$ and $y$ are variables. Here, let me give you an example showing there exist such questions in ...
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complexity of $\ {n \choose n/3}$

I know that the complexity of this combination $\ {n \choose n/3}$ is of $\theta(n^{n/3})$ , but I'm in need of help proving it.
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Help me to find mixed chinese postman problem (MCPP) complexity

I know that MCPP is NP-Complete. Also, I have problem formulation: Chinese postman problem for mixed graphs. I was given a task to evaluate the number of operations required for a complete re-election ...
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Big O for factorials

Hello I have trouble proving:$$(n+1)!\notin O(n!)$$ My first step is the following: $$(n+1)!-cn!\le0$$ Can you please help me with the next step?
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Computational complexity of solving linear diophantine equations?

Is there any good complexity upper bound for checking satisfiability of a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$? I found some estimate on computing the Smith Normal Form $N$ such that $...
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What does $\mathbb{Z}[X]$ for a polynom mean?

I have a proof saying that a Polynom $p \in \mathbb{Z}[X_1,...,X_m]$ I'm a bit confused of this notation because neither X nor the m is explains somewhere. Does somebody of you know the notation?
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Time taken to run a function R(n,a)

R(n, a){ if n = 1 return(a); if n > 1 return (R(n − 1) + R(n − 1) + 1); } Could you please explain me why the estimated time taken to run R(n, a) as a function of n is: (2^(n−1))*(a + 1) − 1 ? ...
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How can I solve this Big $O$ exercise? [duplicate]

How can I prove that $n \log_2(n) ∈ O(log(n!))$ is true? We start by supposing that $f(n)< c g(n)$ is true, which means that $n \log_2(n) > c \log(n!)$ for all $n>n_0$ and $c>0$.
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What is the Order (Big O) of this polynomial?

$$\frac{2n^{14} + 7 n^8 - 3}{3n^8 - n^4 + 3}$$ If this division is $p(n)$, I have to write $p(n) = O(n^k)$ I guess the answer is $n^6$, but how can i solve it step by step?
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Turing Machine problem for unary division

How do I design a turing machine for this? Divide a given number by two in the unary number system.The quotient and the remainder should be written on the tape separated by a blank.
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Prove little-o example [duplicate]

Let $f(x)=\log x$, and $g(x)=x^i$, where $0<i<1$. How can I correctly proof that $f(x)=o(g(x))$? Try 1: By the definition of little-o, a function is little-o of other function if $|f(x)|\leq C|...
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How to prove that $f(x)$ is $O(x^i)$ for a general polynomial

Let $f(x)=a_ix^i + a_{i-1}x^{i-1} + \ldots + a_0$ where $a_i>0$. How can I proof that this general polynomial with real coeficients is $O(x^i)$ using the Big-O notation theory. Try 1: I thought ...
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Practical example of superiority of randomized algorithm

I'm looking for an example to show my students of an algorithm for which randomization of some kind leads to better performance on average. And I don't want that randomization to be of the Monte Carlo ...
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Graham's Scan Polar coordinates

I have this set of six cartesian coordinates and i need to sort them in order to apply Graham's. But i can't figure out how to do that. P[0] = (1,1) , P[1] = (2,6), P[2] = (3,3), P[3] = (4, 2), P[4] = ...
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Sieve of Eratosthenes Time Complexity Clarification

I've found plenty of sources claiming that the time complexity of the prime sieving algorithm Sieve of Eratosthenes is $O(n\log(\log n))$ where $n$ is the input. However, is this $\log_{10}$ or $\ln$? ...
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How we can compute complexity of algorithms in general?

I want to know how compute the complexity of below algorithm : Let $N$ be a positive integer that we don't know it's decomposition. Let $N$ has divisor $b$ such that $b\geq N^\beta$ , $0\leq \beta \...
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relations between Lower bound of 2 algorithems

I am given two algorithms A and B, with worst time complexity $$ f_A (n) $$ and $$f_B (n)$$ Respectively.Now it is given that: For each n there exists and input x of size n such that the number ...
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prove that a polynomial is lower bounded

I need help with this question from Data-Structure course. I need to prove that the following polynomial is lower bounded by $n^k $, meaning I need to show that: $$ p(n) = b_kn^k - b_{k-1}n^{k-1}-...
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Minimal Elements with respect to big Oh

Let $\mathcal{F}$ be a finite set of functions from the natural numbers to the natural numbers. Consider the set $S_{\mathcal{F}}=\{g:\mathbb{N}\to\mathbb{N}\mid f\in O(g)\text{ for every } f\in\...
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At what point does exponential growth dominate polynomial growth?

It's well-known that exponential growth eventually overtakes polynomial growth (link, link). So for any non-negative integer $d$ and positive $\epsilon$, there exists $t^* \ge 0$ for which $$ 1 + ...
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Looking for info on representation of a diophantine equation as system of equations over finite field/boolean algebra

Suppose that $x$ is a positive integer. Fix some prime $p$. Then there exists some non-negative integer, $L$, and $\{x_0, x_1, . . . , x_L\} \subseteq \{0,1,...,p-1\}$ such that, $$x = \sum_{n=0}^{L}...
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38 views

MATLAB “back slash” computation [closed]

I am looking at a MATLAB code that times the backslash operator for several cases. I will list the cases below: Note: all of these are for m = 5000 1) Z = randn(m,m); A = Z'*Z; b = randn(m,1); tic; ...
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How many arithematic operations(flops) are to $n×(n+1)$ matrix of system?

Source: Linear Algebra and Its Applications David C. Lay A system of n equations in n unknows correspond to $n×(n+1)$ augmented matrix. One book says the reduction(elimination) to echelon form can ...
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Hardness of approximation for linear equations

Given a system of linear equations in n variables with coeffcients that are rational numbers, determine the largest subset of equations that are simultaneously satisable. Show that there is a ...
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Fast algorithm to recognize sortable sequences

Every sequence is sortable in the worst-case by a $O(n^2)$. However, if we restrict sorting primitive, we get an interesting problem. I am interested in this sorting problem: Input: a sequence $A$ of ...
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Good books on Algorithms for a math major without any programming experience?

I couldn't find this question anywhere else so it may not be apt. I am an undergraduate mathematics major and during my discrete math class I really enjoyed the study of algorithms and recursive ...
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Prove that an upper bound is incorrect

Probably a simple question that I cant figure out from data structure course: I need to disprove the following statement: $$ 8n^3 + 12n + 3\log^3n \ge n^4 $$ Now I know that from some value $n_0\in\...
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(Complexity) Subset of set of premises and the entailment problem

I've a finite set of propositional formulas $\Gamma$ and a logical conclusion $\psi$ over variables $X$. The following decision problem arises: Does a cosistent subset $\Gamma' \subseteq \Gamma$ exist ...
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Determine number of vertex in a graph

I'm trying to determine the number of vertex in a graph $G=(V,H)$ where: $\displaystyle V = \left\{ v = (x_1, x_2, x_3) \in \mathbb{Z}_p^3:\sum_{i=1}^3x_i\equiv0\right\}$ with $p \ge 3$. Equivalent ...
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Does ray tracing have any speed ups in algorithm running time in the frequency domain?

Could ray tracing be Fourier-transformed so that all calculations are done in the frequency domain? I think ray-tracing a set of rays $S$ from the eye into the view frustum might be more efficient ...
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What is the most efficient algorithm for factorisation when an approximate value of one factor is known

If I am given the following number: 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350 692006139 And am told that one of ...
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Find a function f(n) such that T(n) is $\Theta(n \cdot log(n)) $

Find a function f(n) such that $ T(n)=16 \cdot T(\frac{n}{4}) + f(n) = \Theta(n \cdot log(n)) $ Also, another section of the question is where $T(n) = \Theta(n^{2})$ I've tried using the master ...
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Parallel Computing Mathematical Question (Amdahl's Law)

I have the following question, and I'm not even sure how to start this: A serial program takes T (e.g. ) hours to complete its execution. Assuming that the interprocess communication in a parallel ...
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Is it faster to calculate inverses of symmetric matrices as opposed to asymmetric matrices? How?

I know there are several methods to inverse or decompose matrices. I am looking for a comparison of the computational cost of inverting an arbitrary real, symmetric matrix vs a real, asymmetric one. ...
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Computational Complexity for lower triangular matrices

I am trying to find the complexity $$ l_{ij} = \frac{1}{l_{jj}} \left(a_{ij} - \sum l_{jk} l_{ik} \right). $$ I have considered $a_{ij} - \sum \ldots$ as being one operation, $l_{jk} \times l_{ij}$ ...
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Complexity of FFT algorithms (Cooley-Tukey, Bluestein, Prime-factor)

I need to be able to explain the complexity of three Fast Fourier Transform algorithms: Cooley-Tukey's, Bluestein's and Prime-factor algorithm. Unfortunatelly, I'm a little lost in the process. ...
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How does one interpret statements like: “The traveling salesman problem is NP-complete?”

The world abounds with statements like: The traveling salesman problem is NP-complete. But when I follow try to follow the Internet's links "down the rabbit hole," I don't get a truly sensible ...
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Relationship between Complexity and Computability

As a response to comments,i'd like to put it in an abstract way,hoping this will make things clearer: f is a well-defined function of countably many inputs:f(a1,...,an,...). For a set of n objects {a1,...
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Undecidable problems involving elementary functions

I am reading the article "Some undecidable problems involving elementary functions of a real variable" by Daniel Richardson and have some problems with understanding Lemma Three : Let $h(w)=w\sin w, ...
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Prove that for every function $s:\mathbb N\to \mathbb N$ with the following constraints holds that:

Hello guys I'm studying Computational Complexity and I have stumbled upon the following question which I has no idea how to even start proving. I would appreciate any help. Prove that for every ...
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Why these algorithms have a linear complexity function?

Considering the following algorithms: ...
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What are the minimum hard constraints that cause the nurse scheduling problem to be NP-complete?

A client wishes to simplify the nurse scheduling problem to 'bypass' the NP-complete nature of this problem. He is hoping to do so by removing the requirement that there are any constraints for any ...
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When is $(mn+1)/(n-m)$ an integer?

For an integer $n$ I would like to find all integers $m$ with $n/2<m<n$ and $$ \frac{mn+1}{n-m} $$ an integer, that is, $$ mn\equiv-1\pmod{n-m}. $$ How can I find these $m$? I could just check ...
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Complexity of recurrence equation 6

$B(n)=B(⌈ n/\log_2 n⌉)+\theta(n)$ B(2)=1 Here is my attempt: \begin{align*} B(n) &= 3B(\lceil n/\log_2 n\rceil) + \Theta(n)\\ &= 3\big(3B(\lceil n/(\log_2 n)^2\rceil) + \Theta(n)\big) + \...
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Complexity analysis of a polynomial and a logarithmic exponential function

I need to find the asymptotic relationship between the functions $f(n) = n^{100}$ and $g(n) = (log_2n)^{(1/2) \cdot log_2n}$. I did the following to show that $f(n) = O(g(n))$: $n^{100} \leq (...