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

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Is factoring polynomials as hard as factoring integers?

There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard? One thing I ...
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622 views

What is the significance of the graph isomorphism problem?

It seems that graph isomorphism is an overwhelmingly interesting problem, particularly computationally. Why is that? What are the (theoretical and practical) implication of the existence of an ...
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219 views

Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n)=4T\left(\frac{n}{2}\right)+n^2$$ My guess is $T(n)$ is $\Theta (n\log n)$ (and I am sure about it ...
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27 views

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 $f : X \to Y$, ...
26
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$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
7
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810 views

Divisor summatory function for squares

The Divisor summatory function is a function that is a sum over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor ...
6
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2answers
2k views

Minimum distance of a binary linear code

I need to find parameters $n$, $k$ and $d$ of a binary linear code from its Generator Matrix. How can I find parameter $d$ efficiently? I know the method that compute all the codewords and take ...
8
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1answer
126 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
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1answer
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 ...
0
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1answer
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 ...
6
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1answer
755 views

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 ...
6
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1answer
481 views

Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?

I have been reading Introduction to Algorithms by Cormen et al. Before explaining Strassen algorithm the book says this: Strassen’s algorithm is not at all obvious. (This might be the biggest ...
4
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1answer
3k views

Upper bound for $T(n) = T(n - 1) + T(n/2) + n$ with recursion-tree

I'm reading through Introduction to Algorithms, 3rd ed. and I got stuck on the following recurrence (exercise 4.4-5): $$T(n) = T(n - 1) + T(n/2) + n$$ The exercise asks you to find the upper bound ...
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2answers
138 views

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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2answers
118 views

How can the following language be determined in polynomial time

I'd love your help with understanding why the following is decidable and can be determinate in polynomial time ($L \in P$). $L=\{(\langle M \rangle,w)|M$ is a Turing machine with Q states and one ...
0
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2answers
275 views

List of calculation rules for asymptotic notation?

Background: I am working my way through CLR/CLRS's proof of the master theorem (section 4.4 in the 1st and 2nd editions of Introduction to Algorithms), and I'm doing my own write-up of this proof1 ...
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62 views

Complexity classes and number of problems

Will every complexity class contain infinite number of problems? If they do not, do common complexity classes (e.g. P,NP,PSPACE,EXPTIME,EXPSPACE etc.) contain infinite number of problems?
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123 views

Determining computational complexity of stochastic processes

I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov ...
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4answers
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Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
9
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1answer
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Computational complexity of least square regression operation

In a least square regression algorithm, I have to do the following operations to compute regression coefficients: Matrix multiplication, complexity: $O(C^2N)$ Matrix inversion, complexity: ...
10
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282 views

Is there any infinite set of primes for which membership can be decided quickly?

The AKS algorithm decides whether or not $n$ is prime in time $\tilde{O}((\log{n})^6)$. I am wondering if there is any faster algorithm to determine membership in some infinite set of primes. What I ...
8
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190 views

Applications of computation on very large groups

I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all ...
6
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1answer
2k views

Definition of time-constructible function

What would be an intuitive notion of time-constructible functions ? Is there a function which is not time-constructible? In my own words I would say a function is time-constructible when 1. it is ...
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11k views

What is the 3SAT problem?

I don't get the 3SAT problem. Can someone explain the 3SAT problem as if I were 5 years old, ideally with examples? Thanks!
6
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322 views

How complicated is the set of tautologies?

Consider the set $\mathcal T$ of all tautologies in the propositional calculus in which the only operators allowed are $\to$ and $\neg$, and involving only the two variables $x$ and $y$. How ...
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3answers
189 views

Using $O(n)$ to determine limits of form $1^{\infty},\frac{0}{0},0\times\infty,{\infty}^0,0^0$?

Is it sufficient to use $O(n)$ repeatedly on $1^{\infty},\frac{0}{0},0\times\infty,{\infty}^0,0^0$ to get determinate forms? For example if we look at $\frac{0}{0}$ then $$\frac{O(f(n))}{O(g(n))}$$ ...
8
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1answer
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Complexity of counting the number of triangles of a graph

The trivial approach of counting the number of triangles in a simple graph $G$ of order $n$ is to check for every triple $(x,y,z) \in {V(G)\choose 3}$ if $x,y,z$ forms a triangle. This procedure ...
6
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1answer
947 views

Why are Hornsat, 3sat and 2sat not equivalent?

I have been reading a little bit about complexity theory recently, and I'm having a bit of a stumbling block. The horn satisfiability problem is solvable in linear time, but the boolean satisfiability ...
3
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1answer
131 views

Confusion related to the definition of NP problems

I have this confusion related to the definition of NP problems. According to wikipedia Intuitively, NP is the set of all decision problems for which the instances where the answer is "yes" have ...
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1answer
65 views

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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1answer
361 views

Polynomial complexity algorithm of partition problem with sets of equal size

Partition problem is well known ( http://en.wikipedia.org/wiki/Partition_problem ). Let's add an additional condition: sizes of both sets should be equal. Is there a pseudo-polynomial solution to ...
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374 views

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so ...
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144 views

How do I grade the complexity of the below math puzzle game?

The game (I built it, and it is currently live on mobile) involves solving a pascal's triangle like grid of numbers with operators between the numbers - an example with 3 rows is: ...
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70 views

Sequences and Languages

Let $U$ be the following language. A string $s$ is in $U$ if it can be written as: $s = 1^{a_1}01^{a_2}0 ... 1^{a_n}01^b$, where $a_1,..., a_n$ are positive integers such that there is a 0-1 ...
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Can all programs be modeled as operations of elementary arithmetic operations on inputs?

In mathematics and computabiltiy theory, we treat all inputs and intermediate results and final outputs as natural number. While algorithms/programs themselves are considered natural numbers, here we ...
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1answer
285 views

Can a Pratt certificate for a prime be found in polynomial time?

Can a Pratt certificate for a prime be found in polynomial time? I guess this is the same as asking whether the AKS primality test provides extra information that allows $p-1$ to be factored quickly. ...
5
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1answer
52 views

Choosing primes uniformly at random

I'm interested in efficient methods of generating prime numbers in a given range [a, b] (or with a given number of bits/digits, etc.). By "efficient" I mean minimizing time, randomness, and perhaps ...
5
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238 views

NP-complete: One proof to rule them all

To prove a decision problem $C$ is in NP-complete, 2 things need to be shown: There is a polynomial verification for $C$ solution. Every problem in NP is reducible to $C$ - You can solve all the ...
5
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1answer
179 views

Primality Testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

What is known about the computational complexity of primality testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ where $d$ is a square-free number? For what values of $d$ is primality testing easy ...
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350 views

Is there a log-space algorithm for divisibility?

Is there an algorithm to test divisibility in space $O(\log n)$, or even in space $O(\log(n)^k)$ for some $k$? Given a pair of integers $(a, b)$, the algorithm should return TRUE if $b$ is divisible ...
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1answer
245 views

Is there any decidable problem that is NOT NP-HARD?

Is there a proof that there exists a decidable problem that is NOT NP-HARD??
4
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1answer
489 views

Question about the simplex method complexity

So I know that in general the simplex method for linear and convex quadratic programming can require exponential time. But assuming a positive semidefinite quadratic program that is solvable by the ...
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2answers
408 views

Asymptotically optimal algorithms

Suppose one has an algorithm to solve a problem using at most f(n) computations with size of input n. How to prove, if such is the case, that this algorithm is the fastest possible for solving this ...
2
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1answer
397 views

Form or asymptotic behaviour of $T(n) =2T(n-1)+n$

$T(n) =$ if $n=1$, then time execution is $1$, if $n \geq 2$ then $2T(n-1)+n$ The options are: $T(n) = 2^{n+1} - n - 2$ $T(n) = O(n2^n)$ $T(n) = \Omega(n)$ $T(n) = \theta(2^n)$ Thanks.
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Algorithm to multiply nimbers

Let $a,b$ be nimbers. Is there an efficient algorithm to calculate $a*b$, the nim-product of $a$ and $b$? The following rule seems like it could be helpful: $$ 2^{2^m} * 2^{2^n} = \begin{cases} ...
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1answer
116 views

Solving recurrence relation: f(n) = 3f(n/2) - 2f(n/4) | f(2) = 5, f(1) = 3

f(n) = 3f(n/2) - 2f(n/4) | f(2) = 5, f(1) = 3 I have attempted to solve it by letting n = 2k f(2k) = 3f(2k-1) - 2f(2k-2) Then set S(k) = f(2k) S(k) = 3*S(k-1) - 2*S(k-2) ...
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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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1answer
60 views

The history function preserves recursiveness

Starting with an effective coding of the lists of numbers, I recently proved that concatenation of lists is primitive recursive. On the way I used that if a function is primitive recursive, then its ...
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1answer
187 views

Algorithm for topological sorting without explicit edge list

Suppose I have a set of vertices $V$ and a function $f(V_1, V_2)$ which given two vertices returns +1 if there is an edge from $V_1$ to $V_2$, -1 if there is an edge from $V_2$ to $V_1$, and 0 ...
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97 views

The use of master theorem appriopriately

I have a recurrence relation and trying to use master theorem to solve it. The recurrene relation is: $$T(n) = 3T\left(\tfrac n5\right) + \sqrt n$$ Can i use the master theorem in that relation? If ...