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

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Betweenness Centrality: How Long Does Mathematica Take?

A simple maths problem. If I have a disk of unit radius and place within it $N$ nodes such that the node density $\rho$ is given by $$\rho=\frac{N}{\pi R^{2}}=\frac{N}{\pi}$$ then connect each node ...
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$NP^{PP} vs. PP^{NP}$, which one subsumes the other?

I understand why P with an NP oracle ($P^{NP}$) subsumes $NP$: because it contains co-NP. But how about NP with a P oracle? Can it be any different from NP? (I'm guessing they are the same otherwise ...
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263 views

Proving Hamiltonian Cycle is NP Complete

I'm trying to learn Complexity classes.I want to show Hamiltonian cycle is NP Complete. The text tells me that Inorder to prove NP-Completeness we first show it belongs to NP,by taking a ...
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72 views

GCD of high order polynomials(modulo large prime)

I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...
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66 views

sorting young's tableau better than n^3

Young's tableau takes $O(n^3)$ to sort. By sorting I mean sort numbers in existing young's tableau. http://en.wikipedia.org/wiki/Young_tableau Simply it is a matrix sorted by rows and columns. ...
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46 views

Formulating the Twin Prime Conjecture as a Language Recognition problem.

I'm trying to figure out how to formulate the Twin Prime Conjecture as a language recognition problem. I've got: A = {p: p is the largest prime such that p + 2 is prime} B = {p: p and p+2 are both ...
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160 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
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99 views

State-Space Complexity of RISK board game

I want to calculate the (state-space) complexity of the RISK board game. Online I found a thesis that outlines that complexity (page 34). Here is the summary: Let M denote the maximum number of ...
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37 views

Understanding an algorithm

I want to understand the above algorithm. My solution says that the algorithm should return $0$ if $n$ is a prime or 1. Otherwise it returns the smallest (positive) non-trivial divisor. Lets ...
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66 views

Pumping lemma proof of $L = \{a^nb^m \mid 0\leq n<m\}$

Prove the following language is not regular using the pummping lemma $L = \{a^nb^m \mid 0\leq n<m\}$ I tried solving this problem what I don't think I was able to reach an accurate proof. But this ...
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Transform a k-CNF formulae to conjunctions of boolean literals

The question comes from Mehryar Mohri's Foundations of Machine Learning. In Example 2.5 the book transform a $k$-CNF formula to conjunctions of boolean literals, but I can't understand the trick in ...
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Theoretical computer science text for mathematician

I am a high school student, I know some basic programming in java,python and visual basic. I love combinatorics and I have encountered various cases in which I have found some problems are really ...
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38 views

Trying to understand the math in a neuroscience article by Karl Friston

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
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65 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
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Analysis of algorithm about complexity

$n$ is $O(\log n)^{\log n}$ ? This is true or false, Give the reasons behind that ? I dont get understand about that $O(\log n)^{\log n}$.
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How to find the asymptotic behavior of these sums?

Let $$X(n) = \displaystyle\sum_{k=1}^{n}\dfrac{1}{k}.$$ $$Y(n) = \displaystyle\sum_{k=1}^{n}k^{1/k}.$$ $$Z(n) = \displaystyle\sum_{k=1}^{n}k^{k}.$$ For the first, I don't have a formal proof but I ...
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62 views

Computable Set & Function

we know that i read this sentence are true? can anyone say an example for following sentence? there are a non computable set A such that
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Solving the equation $n\log n = 10^9$

This seems very basic (I guess my calculus needs brushing up). Is there a way to find n without a calculator in this one? $10^{9} = n\log(n)$ My Attempt (log is base 2 base on the book convention.) ...
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How long would it take to test all possible states of a 64x64x64 cube of bits?

Imagining a solid cube of $64 \times 64 \times 64$ bits (each of which can have exactly two states), how long would it take to test all possible states of one of these? Let's also assume we're using ...
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77 views

TAUTOLOGIES NP-Complete Condition

The decision problem TAUTOLOGIES is, Given $\forall x_1 \forall x_2 ... \forall x_n$ $\phi(x_1, x_2, ... x_n)$ a set of universally quantified Boolean variables and a Boolean formula ...
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33 views

Splitting a graph into two isomorphic parts

Say a graph $G$ has $2n$ vertices. I'd like to know if I can partition the vertices of $G$ into two parts $X$ and $Y$ such that $G[X]$ is isomorphic to $G[Y]$ ($G[S]$ denotes the subgraph of $G$ ...
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Linear-bounded Turing Machines

What is the class that contains linear-bounded Turing machines? Is it possible to diagonalize on this class? And is there any Universal linear-bounded Turing machine that can simulate linear-bounded ...
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19 views

$P=NP \Rightarrow P(S)=NP(S)$?

If $P=NP$, then is it possible to conclude that for every sparse set $S$, $P(S)=NP(S)$? ($P(S)$ means a class of sets for which a Polynomial Deterministic Turing Decider with Oracle Set $S$ exists, ...
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Hardness of bounded modular square root of 1

If we know any square root of 1 modulo N different from 1 and N-1, then we can find a nontrivial factor of N. So to find such a square root has a certain hardness. In fact, if in general we ask to ...
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math rules when having 2 variables in Big-O

I came across the following in some lecture notes: O(log n) + O(log m) = O(log n + log m ) = O(log (m + n)) that last step to ...
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Integer programming feasibility is NP-what

What is the complexity class of the general problem of integer programming feasibility? The sources I've looked at are, in my opinion, very confusing. Some say NP-hard, some say NP-complete. Some ...
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36 views

How on earth will anyone prove $n^3-3n^2+n-1=Θ(n^3)$

I know its homework question.Sorry for that.But i was solving all problems of Skiena and chapter and got stuck to this problem of 2nd chapter 2.10. Its easy to prove $n^3-3n^2+n-1=O(n^3)$ because ...
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28 views

What's the complexity of expanding a general polynomial?

Suppose I have a polynomial in the form $(a_1 x_1+ a_2 x_2+...+ a_m x_m)^n$, where $x_1,...,x_m$ are the independent variables. I want to expand it to the form of sum of products. What is the ...
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49 views

Recursive Set and Complement Problem

if we have $$A=\{x:|W_x\ne\phi\}$$ can we say always my tight listed below is true? $A$ is recursive , $A$ is r.e, complement of $A$ is r.e, complement of $A$ is not recursive?
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Double sigma summation is in complexity calculation

Basically i was reading skiena and doing exercise of 2nd chapter.The result of my calculation got differed from the actual solution given on Solution site and there is one thing i don't understand how ...
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Confused with the definition of NP

In the blog post at: Concrete Nonsense I read: NP is the set of problems that can be solved by polynomial-time non-deterministic algorithms. An equivalent definition of NP is the set of ...
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31 views

Using the pumping lemma to prove that a certain language is not regular

I've been trying to understand the pumping lemma since forever, I just don't know what it does, I have no clue what any of it does. My college professor sucks, he thinks writing a bunch of stuff on a ...
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48 views

What is the time complexity of an $O((\ln n)^{\ln n})$ algorithm?

How can the time complexity of an $O((\ln n)^{\ln n})$ algorithm be simplified and compared to some other time complexities?
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Reduction between languages in P

I have a simple question about the class P: Is there exist a polynomial time reduction between every two languages A, B in P?
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Simple question on complexity

I just started to learn the complexity theory, and I have a simple question: Let's say that there's a language B in NP, such that there's no polynomial reduction from B to a given language A (which ...
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Trying to prove something in complexity

I just started to learn about complexity-theory, and I'm trying to prove this: If P=NP, then every (non-trivial) language in P is NP-complete. Can someone give me a solution please?
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Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
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Prove in complexity theory

Given a language A, which is in NP and also not NP-complete, I have to prove that P != NP. [Note: A is not trivial] Any suggestions?
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$O(n^{\log(n)}) $ time algorithms

Is $O(n^{\log(n)}) $ time algorithm considered of exponential time ? Is it applicable ?
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Generalization of Jacobi symbol for higher powers?

Let $n$ be an odd positive integer of unknown factorization, and let $x$ be relatively prime to $n$. The Jacobi symbol $\left(\frac{x}{n}\right)$ gives me partial information on whether $x$ is a ...
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Find a simple Theta bound for a geometric series

I'm working on a question like: find a simple g(n) (big theta) for $ f(n) = \sum _{i=1}^n 5^i $ My working starts with this $\frac{5-5^n}{1-5}$ which is not equivalent to the correct answer from ...
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Complexity class notation

In my CS courses we often use Big-O notation to denote the complexity of a certain calculation. However, we often also write stuff like: $$mO(1) = O(m),$$ or: $$O(m) + O(n) = O(m+n) = ...
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Computational complexity of numerical integration of gaussian function

$ \int^{b}_{a} \exp(-x^2)\,dx$. I have the following two questions regarding the above integral expression of the Gaussian function: Is there a numerical method we can use to solve the above ...
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350 views

Calculating time complexity of algorithms written in pseudocode.

Nowadays we are interested to find some algorithms with a prescribed running time. For example if for certain decisional problem $X$ there is an algorithm with running time $O(n^3)$ we try to break ...
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Problem on time complexity

If $a = O(m)$ and $b = O(n)$, is it true that $a+b = O(m+n)$? I would try to break it down to $a \le cm$ for some $c$ and $b \le dn$ for some $d$, so if I were to add the right hand side, it would be ...
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Is there a fast algorithm for computing the $(2^n)+1$ th last digit of $3^{2^n}$ in base $2$?

Is there an algorithm such that for some polynomial p, it always computes the $(2^n)+1$ th last digit of $3^{2^n}$ in base $2$ in at most p(n) steps for all nonnegative integers n? I'm only asking if ...
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Mixed Q horn SAT

I am familiar with Horn formula: Formula whose clauses have atmost one positive literal. I am also familiar with Mixed Horn formula: Formula whose clauses are either 2 CNF or Horn. Question 1: But, ...
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Multiplying Big-Os

I've seen the following in a text: $\mathcal{O}(\sqrt{n})\cdot \mathcal{O}(n\log n)$ How is that even defined? Ok, I guess one can replace it with: $\mathcal{O}(\sqrt{n} \, n\log n)$ Is that ...
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Find subset sum problem verifier and its complexity

As homework we need to find a P-verifier for the subset sum problem. Given: natural numbers $a_1, \cdots, a_n$ and $b$ Output: YES if there is a subset $S \subseteq \{a_1,\cdots,a_n\}$ where ...
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Master Theorem and Logarithms

I am really struggling to understand the Master Theorem when in a formula: T(n) = aT(n/2) + f(n) f(n) takes the form of log(n), nlog(n) or nlog^k(k) For ...