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

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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 - ...
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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 } ...
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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 = ...
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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 ...
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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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42 views

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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4answers
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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 ...
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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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1answer
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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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1answer
14 views

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 ...
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Undecidable problems involving elementary functions

I am reading the article "Some undecidable problems involving elementary functions of a rial 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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40 views

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 ...
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Calculating time complexity of an algorithm.

In the chapter Algorithm Analysis of the book Algorithm Design Manual there is an example of string matching algorithm, I am typing the partial code below: ...
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1answer
72 views

Is cracking MD5 hash a form of P VS NP problem? [closed]

I have a question,Is cracking MD5 hash a form of P VS NP problem?
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32 views

Understanding of what it means to say that a question is in NP

Let's say the the function $f$ can be evaluated in polytime in the size of the input $x$. Are the following problems in NP? Is there an $x$ such that $f(x) = y$ for a particular value of $y$? Find ...
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The role of the extraction matrix in a Kalman filter

The extraction matrix shown as $H_k$ below, transforms the state vector into a form that can be subtracted from the measurements vector: $\hat{X}_k = \hat{X}_k^- + K_k ({z}_k - H_k \hat{X}_k^-)$ ...
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Using Newton Binomials and Combinatorics to reach this big O result?

I'm trying to understand this theorem proof: Theorem. Given a set of n agents, the dynamic programming algorithm, DP, computes an optimal coalition structure in $O(3^n)$ time. Proof of theorem How ...
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What is computational complexity of $Ax=b$ when size of A increasing

I have a linear equation $$Ax=b$$ where $A$ is non-singular matrix $N \times N$, $x,b$ are vector $N\times 1$, $A,b$ are given and I want to find $x$ It is clear that $x$ can find by $x=A^{-1}b$. ...
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Finding All Combinations of a Hierarchical list Where Conditions Are Involved

I want to find all possible combinations of a list that looks like this. a) Option 1 Sub Option 1 b) Option 2 Sub Option 2 c) Option 3 The catch is that there are some simple and some ...
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46 views

Average time complexity on finding all common substring of a string

Background Information & Research I'm working on an algorithm where you have to find all common substrings for a given string. For instance find("ABC", "ABCD") would result in {A, AB, ABC, B, BC, ...
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102 views

Euclid's algorithm and Gaussian elimination: most computationally efficient approach

I think this is more a mathematics question than a computer science one - but as it is about computational complexity it could be either, so apologies if you think it is in the wrong place... The ...
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On a subset sum version.

In subset sum we ask 'Given $n$ numbers in $\Bbb Z$, is there a subset of them that sums to $0$?' this is $NP$ complete. Consider variant: 'Given $n$ of degree at most $d$ polynomials in $\Bbb Z[x]$ ...
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Algorithmic time complexity of Newton's method vs bisection method

In the context of root finding, it is often stated that the bisection method is slower than Newton's method due to linear convergence. However, I am trying to understand why this is the case from an ...
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Why does the cutting plane method for integer programming run in exponential time?

I am looking for a proof of the fact that the cutting plane algorithm for integer programming does not run in polynomial time. The algorithm consists in adding constraints to the initial problem in ...
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Test symmetricity for a sparse matrix

I have a sparse matrix in LIL (List of Lists) format. I want to test whether the matrix is symmetric or not. Let's say I have n non-default elements. What's the ...
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Complexity Of Recognising Complete Multipartite Graphs

Short question: Is there a linear time algorithm for recognising complete multipartite graphs?
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Need an example shows why SAT is NP problem

Kindly, I have two questions: (1) Are NP-hard, NP-problem, and NP-Complete are just synonyms of each other? (2) I understand that SAT is NP problem that cannot be solved in polynomial time ...
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Reduction to/from REC and RE language?

Let $X$ be a recursive language and $Y$ be a recursively enumerable but not recursive language. Let $W$ and $Z$ be two languages such that $\overline{Y}$ reduces to $W$, and $Z$ reduces to ...
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Complexity of Discrete Mathematics algorithms

I'm new to decision maths and searching algorithms, but one thing I don't understand is how it's determined what complexity (in big-O notation) an algorithm is? For example, I've seen $O(2^n), ...