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

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I want to know an estimate of $a_{i, j}$

Let $$ a_{ij} = \begin{cases} -1, & \text{if $i = -1$ and $j = -1$} \\ 1, & \text{if $i = -1$ and $j \ne -1$} \\ 1, & \text{if $i \ne -1$ and $j = -1$} \\ a_{i-1, j-1} + ...
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Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
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38 views

Computational complexity of expanding a MacLaurin/Taylor Series

What methods exist to computationally determine the first $k$ coefficients of a function (possibly polynomial or rational polynomial function)? How do Mathematica/MatLab/Maple/etc. solve this ...
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How do you reckon Big-O analysis with infinite problem sets?

Let $f : X \to Y$ be a problem, for instance, $f: \Bbb{Z} \to $ a factor. Given input measure $n = |x|$, then our problem is $O(g)$ if there exists an algorithm running on a standard machine, and an ...
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My notes on $\Bbb{Z}/p\Bbb{Z}$-theoretic computational complexity

(Question at the very bottom) Def 1. Let $F = \Bbb{Z}_p$ be a finite field. Then an $F^k$-machine is a machine with $k$ input / output memory slots. All computations are done in the field $F$ and ...
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33 views

How do you define computational complexity abstractly?

Let the problem we're studying be $f : X \to Y$. Say, I don't know what I want to define time-complexity with respect to, I just know I have a map $|\cdot| : X \to \Bbb{R}$, such that $|\cdot| \geq ...
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Not every polynomial in $\Bbb{Z}_p[x]$ can be factored, but can you do next best?

If $f \in R = \Bbb{Z}_p[x]$ is irreducible or doesn't have many factors then it could be hard to compute? Possibly, I'm not saying, but... any way, what if $f = h - g$ where $h, g$ are heavily ...
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Do there exist polynomials not computable in polynomial time?

Motivation: Computing a problem in $k$ memory slots Do there exist polynomials in $R = \Bbb{Z}_p[z_1, \dots, z_k]$ that can't be computed in time polynomial in $k,p$? Thanks... Good luck! Edit. I ...
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Any problem computable in $k$ memory slots can be computed with polynomials.

Let our memory slots be represented by elements of $\Bbb{Z}_p$ for a prime $p$. $k$ memory slots would be $k$ copies of the ring: $R = (\Bbb{Z}_p)^k$. Suppose that for a problem $f : X \to Y$, ...
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24 views

circuits complexity

I ran into this claim and I dont really understand why it is true ( and I need to use it for proving other things) Every function from $(0, 1)^n$ to $(0, 1)$ is computable by a $2^n10n$-sized ...
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Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
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61 views

Existence of a det. poly-time algo for problem $f: X \to Y$.

$f : X \to Y$ is a deterministic polynomial-time algorithm for problem inputs $x \in X$ and problem outputs $f(x) = y \in Y \iff $there exists a polynomial $P_f \in \Bbb{Z}[x_1]$ such that $C\cdot ...
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How do you find a minimum of a function with these tools?

Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$. Is there a polynomial time ...
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34 views

Can cuts of size 2 be detected in linear time in an undirected, unweighted graph?

I'm having trouble finding any literature on the specific subject of 2-edge cut detection. It's not hard to come up with an algorithm that finds all 2-edge cuts in quadratic time, but it's not clear ...
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Complexity of finding extremal rays

Suppose that $\{L_i\}$ is a collection of $k$ linear forms on $\mathbf R^n$. Let $$C=\{x \in \mathbf R^n : L_i \cdot x \geq 0 \text { for all } i \}$$ be the closed convex cone defined by the ...
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If $P=NP$, prove that $L' \in NP$

I think I'm overthinking this problem and need some hints in the right direction. The goal of this question is to show that if $P=NP$ then for every language $L \in NP$ via a polynomial time verifier ...
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101 views

Order functions by speed of their asymptotic growths

We are given list of functions. Task is to sort it by the speed of their asumptotic growth in ascending order. Yes, it's a homework. I already spent some solid amount of time calculating limits. I ...
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125 views

What does psuedo-polynomial algorithm for subset sum problem mean?

Help me out here - just trying to better understand what 'psuedo-polynomial' means... If the input to an NP-Complete problem is 100 items(ie n=100), and the 'target' is the actual value '100'(t=100): ...
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Prove 2-HamiltonianCycle $\in \textbf{NP}$

Just want to verify that I have the right idea here with this hamiltonian cycle question. $HC$ = $\{\langle G \rangle$ | $G=(V,E)$ is an undirected graph such that there is a simple cycle (no vertex ...
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Implication of P =NP on video games?

I was wondering if NP problems were actually solvable in P time, then what will be the impact on Video Games, if any ?
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Is $\log^* (n+1)^{n+2} \in O(\log^* n)$?

I would like to know if $\log^* (n+1)^{n+2} \in O(\log^* n)$, where $\log^*$ is the iterated logarithm. I tried doing: $ \log^* (n+1)^{n+2} =\\ \log^{*}(\log(n+1)^{n+2})-1 =\\ \log^{*}((n+2) \cdot ...
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Examples of functions which grow faster than their computational complexity.

A good example of this would be the function $f$ defined as follows, $f(n) = (10^n-1)$. While in this form it's equation is exponential, it is easy to note that $$f(n) = 99...9 \,(n \text{ times}).$$ ...
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Decidability of normal modal logics

Let's say we have systems of modal logic defined as smallest sets containing propositional tautologies, all instances of schema $\square F \to (\square(F \to G) \to \square G)$ all instances of ...
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exponential time complexity

given that: $f(n) = a^n$ and $g(n) = b^n$ where, a,b are positive integers and n is a positive real number does there exist some $f(n) \notin \mathcal{O}g(n)$? ie. $f(n) \leq c_1 \cdot g(n)$ is ...
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A confusion about RP class of problems

I have some notes which introduces the quantifier $\exists^+x$ and interprets it as "the overwhelming majority of $x$". Then, it defines RP (Randomized Polynomial) as: $$ L\in RP\Leftrightarrow ...
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Why one defines the proper complexity functions?

Definition: A proper complexity function is a function $f$ mapping a natural number to a natural number such that: $f$ is nondecreasing There exists a $k$-string Turing machine $M$ such that on any ...
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113 views

Minimizing Height of a Table

This optimization question popped into my mind while working with latex tables: Suppose we have a table with $m$ rows and $n$ columns, and for each $1\le i\le m,1\le j\le n$ we are given $T(i,j)$ ...
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What is the Big O complexity of this equation?

$$\frac{n+1}{2}(\log_2(n-1))+\frac{n+1}{2}-\log_2(1)-\log_2(3)-\dots-\log_2(n)$$ What is the Big O complexity of this equation? My initial guess is $n \log(n)$ But after a calculation, I ...
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The complexity of counting solutions to $x_1 + \dots + x_m = N$ in non-negative integers under constraints

Consider the equation $$x_1 + \dots + x_m = N$$ where $x_1,\dots,x_m \ge 0$ and under the additional constraints $x_k \le a_k$ for $k=1,2,\dots,m$. I'm interested in knowing whether the number of ...
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Why $17T(n/16) + n \log n$ satisfies the case 2 of the Master Theorem?

Using the Master Theorem, we have that $17T(n/16) + n \log n$ is $\theta(n^{log_{16}17} log^2 n)$ My question is, why $n \log n = \theta(n^{\log_{16}17} \log^1 n)$, being $\log_{16}17$ approximately ...
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114 views

Switching edges and vertices

I am attempting to convert the problem of finding an edge dominating set into a dominating set. I need a way to change the edges of a graph to the vertices and the vertices of a graph to the edges, ...
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Growth rate of $n^{\sin n}$

Is there a way of comparing the growth of functions $ f(n) = n ^ {\sin(n)} $ and $ g(n) = n ^ {1/2} $ in terms of $ O, o, \Omega, \omega, \Theta $ ? Periodically, $ f(n) $ keeps going above and ...
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Lowering the start indexes in sums

I'm implementing a CYK algorithm in my software and I've found a pseudo-code on Wikipedia. Here's its complexity(modified version for special use, which doesn't go from ...
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What is the complexity of computing some problems related to real analysis

I was thinking. I do not know whether my question has any sense. I want to know is there any way to compute analytically or explicitly some of the problems give below. What is the complexity of ...
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problem about computational complexity

Exercise: Prove that the computational complexity of the binomial coefficient \begin{equation*} \binom{m}{n} \end{equation*} is O($m^{2}$$\log^{2}n$). using the fact that the computanional ...
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What makes the permanent lot more difficult than the determinant

The permanent of an $n$-by-$n$ matrix $A$ = $(a_{i,j})$ is defined as: $\operatorname{perm}(A)=\sum\limits_{\sigma\in S_n}\prod\limits_{i=1}^n a_{i,\sigma(i)}$. ...
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Why is Dantzig's solution to the knapsack problem only approximate

For a bunch of items with values $v_i$ and weights $w_i$, and with a total weight $W$ that our bag can carry, how do we achieve maximum total value without breaking the bag? Dantzig proposed that we ...
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Big O/little o true/false

These are all from Sipser's book, second edition. I was just hoping someone could verify/explain those that are more difficult for me. $2n = O(n)$: true $n^2 = O(n)$: false $n^2 = O(n\log^2 n)$: I ...
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How to figure out the time complexity for an algorithm with dynamic input parameters?

I have an algorithm that depending on the length of the input array and its values could take more or less operations to complete, for example, for a set with some length it could take 10.000 ...
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65 views

From programming to mathematics

I'm studying algorithms design and analysis, but there is a code that I can't understand. I know that: Let $\mathcal P$ be the main program, and $\mathcal P \in O\left(\varphi(n)\right)$ with ...
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Minimum number of different clues in a Sudoku

I wonder if there are proper $9\times9$ Sudokus having $7$ or less different clues. I know that $17$ is the minimum number of clues. In most Sudokus there are $1$ to $4$ clues of every number. ...
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Complexity of primenumber test

The german wiki claims that the approach to check if any number before p is a divisor of p is a polynomial time algoritm. I dont understand this claim. Because imho this is linear, which is polynomial ...
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Sum of a sum [algorithm design and analysis]

I'm studying the algorithm analysis of one piece of code, and I have to find the big-O notation of the sum of a sum. ...
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Is there a limit for how “good” a numerical method can be?

Multiplying two matrices $A \cdot B$ of size $n \times n$ in the trivial way requires $n^3$ computations. However, more efficient algorithms such as the Strassen algorithm have a lower complexity of ...
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71 views

Understanding the meaning of the Linear speedup

The linear speedup theorem informally says the following thing: If $M$ is a Turing machine that operates with time $f(n)$ to do a certain task on some input $x$ then for every $\epsilon>0$ we can ...
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Help making the distinction between polynomial and exponential time

I'm trying to understand how problems are categorized in these two classes. I have a specific problem I'm looking at, the directed path problem: PATH = $\{\langle G,s,t \rangle | G$ is a directed ...
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264 views

If the union of two languages is NP-complete, is one of them NP-complete?

Question 1) If $A\cup B$ is NP-complete, and $A$ is NP, and $B$ is P, then is $A$ NP-complete? I don't think so but I am unsure. When I try to reduce $A\cup B$ to $A$, I fail because strings in $B$ ...
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What is the sum of recursive logarithms?

I am trying to deduce the complexity of a rather odd algorithm. I have got it down to this form: $$ O(n \times (\sqrt n)^2 + n \times (\lg \sqrt n)^2 + n \times (\lg \lg \sqrt n)^2 + \space ... + ...
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Why is integer programming in fixed dimension easier than in general?

When the dimension is an a priori fixed constant, then integer programming feasibility (the existence of an integer point in a polyhedron) can be decided in polynomial time. If the dimension is not ...
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Polytime implementation of Discrete Log using primitive recursive functions

The primitive recursive functions are defined by Godel as: $z() = 0$ $s(x) = x+1$ $\pi_i(x_1, \dots, x_k) = x_i$ Plus closure under Composition: $h(x_1, \dots, x_m) = f(g_1(x_1, \dots, x_m), ...